The best use of battery power

Coin cells will not work well with your needs. While it varies by brand and size, most coin cells have low capacity, and an internal resistance that limits current output. A CR2032 has a nominal 250ma capacity (and that varies based on current load btw), and an internal resistance (Equivalent Series Resistance or ESR) of 18 to 30ohms, again based on current load). Depending on the speed of the current draw, you are going to get voltage drops, and quickly drain the coin cell. If you can get that much current out of it at a given time. 70mA is pushing it even for high quality CR2032, and 220mA is improbable, frankly, impossible. That is why your CR2032 boosted to 3.3v isn't working. The current draw of the regulator is causing the cell's voltage to drop.

That is, without having multiple in parallel (or using high capacity caps). That's the second part you asked. Batteries in series are a sum of the voltages, with the same amount of current. Batteries in parallel are a sum of the current capacities, with the same voltage.

As for your options, one or two AAA batteries with a boost converter would suit you better. It gives you relatively high capacity while still keeping space to a minimal. The TPS61200 you mention would work, if you can deal with a tiny (3x3mm leadless package. The TPS6107x family would do the same, same minimal components, in a sot6 package. You only need 1 battery for either of these. An AA gives 2400mAh nominal, an AAA gives 1200mAh nominal.

Well there is also the noise susceptibility of your circuit, which will be lower if you use 5V (you need higher noise levels to disturb a 5V circuit).

Some components have very different analog performance based on the supply voltage, if you use an op-amp the common mode voltage or the voltage difference needed from the rails might be a point to look out for and help decide which is better for you. So even if all your components work at 3.3V and 5V, your circuit might not do what you expect if you change the voltage.

@Passerby mentioned that 5V is a defacto standard due to USB. You have to be aware though, that if you need a stable supply voltage USB is a bad thing to use directly.

USB actually allows for quite a range of voltages on your device, behind a passive hub they can be as low as 4V and can go as high as 5.25V. So in that scenario you are better off using 3.3V with a voltage regulator on your board.

The maximum operating frequency of your controller might, as pointed out, also be dependent on the supply voltage. Though newer controllers run on much lower voltages internally (like 1.2-1.8V) and don't have a dependency between frequency and external voltage anymore (like the STM32 series).

And of course the power consumption, you'll gain quite a lot from going from 5V to 3.3V. It's only using 66% of the power in a static case and only (3.3^2/5^2) = 44% for the dynamic CMOS case (your actual benefit will lie somewhere in between). But if that is actually beneficial (on a global scale it is) or needed depends on your supply situation and the current you need. It could happen that if you go down to 3.3V your supply gets less efficient and the power you saved is wasted in the lower efficiency turning it around (shouldn't but could).

Another valid point mentioned by Paul in a comment below is that if you interface to some legacy devices, you might need the 5V. Some microcontrollers offer so called 5V tolerant pins, but those are only able to accept inputs up to 5V (often only in digital mode). The output is limited to VDD, so there you need a level shifter if the 3.3V are not accepted by the legacy device as high level. Which you can save if you use a native 5V controller.