You apparently have a unintentional radiator, so that means you don't actually have to have it tested. It is your obligation to make sure it meets the emission requirements if you sell it in the United States, but how you decide it does that is up to you. There therefore aren't any specs on what the equipment you use to test for compliance needs to be, since you don't have to test for compliance in the first place.
However, unless this is a very minor niche low volume product from a obscure company, it is a good idea to get it tested. Effectively, you can't do this yourself unless you are willing to invest 100s of k$ in a calibrated setup, and then spend a significant amount annually to keep it calibrated and verified. The test equipment itself is only the beginning. You need a proper range or anachoic chamber, calibrated antennas, etc. From the costs you were mentioning, this route is clearly not for you.
What you need to do, if you want to be really sure your device meets the requirements before you sell it, is to hire someone else to do the testing that has already spent the large sums on a proper setup. There are test houses all around that do exactly that as their main business. There are probably half a dozen within 2 hours drive from Boston, for example. I don't know what it's like in IN, but there are bound to be a few around. If you know what you are doing and have designed your device with emissions in mind, then testing may cost as little as $3-5k. The price goes up from there inversely proportional to how much you know what you are doing.
Your own test equipment will be useless for making absolute measurements, but it can give you some idea which frequencies you have spikes at. If you are careful to create a repeatable setup, then you can use it to test the relative effects of various changes to your design. For example, you might see you have a peak at 327 MHz with a arbitrary value of 20 dB with the way you arranged the receiving antenna, oriented your product, routed the ethernet cord coming out of it, distance from antenna, etc. After adding a few caps and series inductors to the power cord, you find the same setup reads 14 dB at the same peak. That means your design changes reduced that emission by about 6 dB. If you flunked formal testing due to that peak being 2 dB above the limit, you are probably OK.
Remember that your own tests are only for relative signal strengths before and after some changes you are trying to test. Repeatability of the setup is critical to get anything useful at all. When I'm doing this, I usually clear off a area, use a non-metal table, carefully tape down the antenna and the cord leading from it, and usually use tape to mark exactly where the unit under test is placed. I also usually tape down dedicated cords that go to the unit that are used only for the testing. Use other cords when running the unit elsewhere. The point is to not disturb the RF testing setup to that it is as repeatable as possible. This includes where you are while you are taking measurements.
From my experience with this (several products, using various combinations of wireless devices), then if the wireless module you use (be it cellular, Bluetooth, BLE, Wi-Fi or ZigBee or ISM) includes an antenna, and the entire module (radio + antenna) has been tested per FCC requirements (and therefore has an FCC ID stamped on it), then you do not have to do any further testing.
Here is an exmple of the type of module I am referring to. Note the chip antenna on the upper left corner.
Note the one pictured above is pretty much blank. The real one I have on my workbench is covered with text and logos, including an FCC ID and CE mark.
As long as you have only one wireless device on your board, (and there are no other wireless devices on other boards) the person using it in their product should be able to use the FCC ID that is stamped on the module.
But, if you have a module that attaches to a separate antenna, and only the module was certified (or not), then you must test the combination together. Furthermore, if you have more than one radio in the final product, then they must be tested together, even if they have been certified separately.
This is my understanding. You should have it verified by someone else before proceeding.
Best Answer
I can understand your frustration, testing is expensive:
Source: https://hackaday.com/2016/09/19/preparing-your-product-for-the-fcc/
Source: https://www.fcc.gov/oet/ea/rfdevice
The only devices that receive exemptions are these:
Source: FCC CFR 15.101
As stated in the question, your device would not fit in the unintentional radiators exemptions category (15.103) your device is definitely not at 15.113 (Power line carrier system)
It could fit into 15.23 and more on that later:
Source: FCC CFR 15.23
Your device is an unintentional radiator but not one under section 15.102 (CPU boards and power supplies used in personal computers.) Because these are related to PC's and their components, these components usually are designed for operating inside of a metal enclosure, they also have their own standards that need to be tested.
If your device is not exempted, then you'll need in the least verification because you have a CPU in your product (and it above 1.7MHz check the Frequency allocation list\chart). If your device starts messing around with radio device or knocks someones cell out, someone might complain. It's when people complain (if your radiating on their band) that you get into trouble and become the "responsible party" and if your found to be above the limits, then the producer of the product gets a fine. And rightly so, it can take a significant amount of resources to find an offending device.
Source Are Your Company’s Consumer Electronics Exempt from FCC Marketing Regulations?
If it's a prototype then you don't have to worry about testing, but you still have to worry if your device is an unintentional radiator
Source: https://www.sparkfun.com/tutorials/398
So stick in an unlicenced band for extra comfort and design around that.
Edit -Subassembiles
Source: http://www.emcfastpass.com/fcc-rules-kits-subassemblies/
So if you want to go the sub assemblies route, then sell them as sub assemblies for use in another product, and don't do any marketing.