Just to let you know what lies ahead....
If you want to go from making a hand-built breadboard or prototype to actual PCB's, you have a lot of hours and anywhere from several hundred to a few thousand dollars cost in front of you, depending on how much you are willing to do yourself.
Schematic capture and PCB layout
First of all you need to capture your design using some sort of schematic capture program, and then design a PCB. One of the more popular programs is EAGLE, which I use. They have a EAGLE Light version ($49), but it can only be used for schematics with one sheet (any size), two signal layers, and 100x80mm (approx 4"x3") routing area. For any serious work, you need at least the EAGLE Standard version, which costs $747. There are probably other less costly (even free) alternatives. There are lots of others that cost thousands or tens of thousands of dollars. In any case you will have to spend considerable time learning how to use the program.
Or you can pay someone like me to do it for you ($$/hour).
PCB Fabrication
Getting boards made is the next step by a PCB fabricator. The problem here is the NRE (non-recurring engineering) costs. Some board houses treat this as a separate figure, and others built it into their per-board quote. In any case, it is almost never economical to have just a few boards made. You might spend $100 for two boards, and $500 for 25. You need to have really large quantities to get down to just a few dollars per board.
The gotcha is, if you make 25 boards, populate just a couple of them for testing and find they don't work (and there is not an easy fix -- e.g. because you laid out a connector backwards), you might end up throwing away the other 23 blank boards away and you would have been better off just getting two. I have stacks of blank PCB's as evidence of this phenomena.
PCB Assembly
Unless you are willing to build the boards by hand, you will need to have them assembled. Surface mount packages are difficult to deal with. If the board has BGA or QFN packages, you probably won't be able to build them yourself unless you have your own reflow oven.
Getting your first two boards built by an assembly house might cost $500. Whereas getting 25 built might cost $1200. (Once again, the problem here is the NRE costs.) Getting down to just a few dollars per board requires (again) large quantities.
And someone else has already discussed the problem of getting parts.
Make sure you use parts that are readily available -- if both DigiKey and Mouser have hundreds of the part available you should be okay. If instead, they have it in their catalog, but it is currently out-of-stock, try to find something else. If you need some special parts that aren't carried by DigiKey or Mouser, make sure you have a reliable source before incorporating it in your product. (Note: the more unusual parts you use, the more likely you will have to add the part manually to your PCB parts library.)
Custom Cases
Do you want to put your board into a case? If you need to have a custom case designed, that will be a couple thou for the designer using a program like SolidWorks (I don't do that, but can recommend someone who can). If you are going to make just a few cases to begin with, you will probably need to go with rapid prototyping, such as Selective Laser Sintering (SLS). Figure at least $100 per case in small quantities. To get down to a few dollars per case cost, you need to have a custom mold made. NRE time again! Plan on spending $10,000 or more for the mold.
And I won't even start on EMC or EMI testing, since I don't know if it applies to your product.
As you can see from all of this, until you get into production, the cost of the electronic parts is usually not the biggest item on a per board basis. Doing your own assembly for small volumes will save you a lot of money. So it is important to design with that in mind -- no impossible to solder-by-hand parts.
To get really low prices for high-volume, generally you need to go offshore -- China etc. But I would avoid doing so in the beginning.
Do not work with the idea that the GREEN LED and the BLUE LED can be operated without series resistor. As a starter the paralleled LEDs with lower forward voltage will starve those with higher forward voltage. Series resistors for each can even this out and take care of that problem.
Next it is typical problem that BLUE LEDs do not always work well from a supply of just 3.3V. Consider getting your design modified to add a 5V source to supply this LED and still apply series resistors for these LEDs for the reason given above. With the problem as stated you may be able to find some BLUE LEDs that will give you satisfactory brightness from 3.3V but they will be dim in comparison to the others.
Do not second guess the need for the bypass capacitors. Put them in by the power pin pairs of each IC chip.
Note you failed to hook up one of the VCC pins of your MCU. It is essential to connect all power and ground pins of IC chips.
If you want to have a charger for the battery then you should plan to put in a regular power supply circuit for your design that can accept input from the battery or from some AC-DC converter module. When the AC-DC supply is present then have the battery go into charge mode.
Best Answer
If you don't have any in-house engineering resources, or if you want to hire out all of the engineering design work, use a "turnkey contract manufacturing service". This type of service is always more expensive than regular contract manufacturing service where the engineering design work has already been done. Typically these turnkey contract manufacturers also handle regular contract manufacturing, so regardless of how you arrive at a design, they will produce it in volume.
However... this request is not quite as simple as "making a custom PCB based on Arduino Nano to a smaller form factor to fit inside your vaporizer's enclosure", and then producing at full volume. The vaporizer has temperatures hotter than melting point of silicon, so heat management is part of the design requirements.
When the printed circuit board is redesigned to fit the smaller form factor for your application, there will be less copper area, so you may find the board runs hotter than your full-size Arduino Nano prototype board running in open air outside the vaporizer's enclosure. FET manufacturers such as International Rectifier typically use a square inch of open copper area to act as a substrate heat sink. Your application may not be dissipating as much power, but may be operating in a higher ambient temperature environment (inside an enclosure, next to a 200C heat chamber). Be careful to get as low a thermal resistance to ambient (theta-JA) as you can, while staying insulated from the high temperature chamber.
Have you tested the prototype board operating at the temperature you expect inside the case enclosure? I assume the Arduino board itself is not inside the 200C heat chamber, since it would melt. But even so, your prototype board may require industrial- or extended-temperature-grade components (-40C..+85C or 0C..+125C) to withstand operation on a smaller board, inside an insulated enclosure, next to a 200C heat chamber. Not only the main ICs but also the connectors, capacitors, resistors, solder, wire, even the PCB material itself must be checked for high-temperature operation in a bill-of-materials review. This is not the most exciting part of engineering work, but it does need to be done.
You don't mention how you manage the thermocouple cold junction compensation in your design. This could behave differently once the prototype board is trimmed to the new size and operating inside the enclosure. Any errors in cold junction compensation will affect the absolute accuracy of the reported temperature.
You don't mention how much temperature hysteresis or error band is needed for the 200C green indicator light. It's also unclear to me whether there is an existing control loop regulating the vaporizer's temperature chamber. If not, you may want to consider bringing out "too hot" / "too cold" signals from your Arduino. Control loops aren't trivial, but since you're already adding a versatile microcontroller board, the incremental development cost would be manageable. You may even think about whether you might want to re-use your new small form-factor custom Arduino board in subsequent vaporizer designs -- do yourself a favor and keep as many spare, uncommitted I/O pins as you can manage. Could be helpful for diagnostics and debugging once inside the enclosure.
The engineering work involved in a project like this is not trivial. You will need someone who can deal with printed circuit board design and thermocouple cold junction compensation. Integrating the board with your product's enclosure is crucial. If you can spare a working vaporizer unit, that would be helpful to an engineering contractor to ensure the board fits and the thermal issues get sorted. There's enough complication I can see already from here; don't plan on taking this straight into high volume production right away. Instead, try building a few more prototypes with small form factor board design, and verify that the performance is satisfactory before going to full production volume. The last thing you want is to blow your budget building 10,000 "prototypes" that don't work.