Yes, you can use pre-approved modules, but your new assembly still requires a spurious emission test (in place of a full emission test). It's a cheaper, faster, simpler test, but approval is still required to sell the new product legally.

The FCC is a massive regulatory moat against hobbyist electronics. You have to make something expensive enough or with enough scale to make enough profit to be able to pay for the expensive testing. If you fail the test and aren't able to fix it on site all of that money will be a waste and that will have been a waste of money.

Also in order to use Bluetooth you will have to pay a few thousand dollars in SIG fees to properly license it without infringing their IP.