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.
Here's my understanding.
Since it is a radio module, then it is definitely an intentional radiator and if they claim it has been FCC-certified it should have an FCC ID on each module. You should contact the manufacturer and ask why it's not there.
If you incorporate the module into your own product, and the module already has an antenna (e.g. a "chip" antenna like the one in the picture) then you don't have to do anything more except provide the same FCC ID on the outside of your product, or (as part of a recent rule change) in an accessible menu if your product has a screen. In any case, the FCC ID must also be included in a label on the outside of the package.
However, if the module has no antenna, and you connect it to one you made (e.g. PCB trace antenna) or bought, then guess what, you have to do the FCC testing all over again(!) which can many thousands of dollars.
It's been a few years since I was directly involved in a product like that (cell phone modem + PCB antenna), so I can't remember if you are supposed to include both the FCC ID for the modem and your own FCC ID, or just the latter. I suspect it's both.
Best Answer
Really the best way to get an answer to questions like these is to call your test house and just ask them, even if you are a first time customer they'll tell you things like this for free.
That said I wouldn't say you are treated as a unintentional radiator per se. You just no longer have to do the intentional radiator testing as long as you are using the same antenna the module vendor used and following their integration guidelines. This should save you plenty of cash on its own.