You need to do it to use any Bluetooth trademark including the logo. If you call the function "BT" then you can probably get away with it, as long as you don't use the Bluetooth rune or any other trademark (like "BLE").
There are potentially patents and IP that you do not pay royalties for if you use a module without paying Bluetooth SIG, but arguably those are a matter for the module manufacturer... so it is a bit of a legal grey area (and the people that will advise you on the realities here are not cheap).
Any intentional radiator will need to go through your market's appropriate certification, but it may be possible to transfer the certification if you use a PCB antenna built into a module and comply with the manufacturer's recommendations there. If you use your own antenna, you almost certainly have to do the RF certification.