Firstly, re. homework help,
you are expected to do at least an initial the design then ask about possible problems; not get others to do the work for you.
For the bluetooth side, a BM62 module is probably a reasonable approach. Bluetooth requires complex, custom integrated circuits so a module with accessible terminals is likely the only possibility. There are many other module types, but a lot of the cheaper ones are data only or clones with very variable reliability and specifications.
FM is quite simple, you add a varactor (varicap) diode to the oscillator; or feed low level audio in to the VCO control on a synthesised transmitter.
Or you can use phase modulation, applied to the signal after the oscillator. With suitable audio preemphasis that produces exactly the same FM output signal.
Re. the transmitter side:
Note that transmitting audio without a very specific licence is illegal pretty much worldwide.
So is using
home-made transmitters; every transmitter type must be certified to not cause interference or use excess power.
You can use extreme low power, a microwatt or so, basically oscillator leakage without any antenna, but that's about the limit of it.
(Commercially made RF modules for such as bluetooth,wifi and other specific defined systems from major manufacturers, are pre-certified to conform to frequency and power regulations).
Having said that - here are a couple of example ham radio transmitter FM VHF designs, that will show some of the concepts involved.
(I noticed a typo on this one - the last paragraph of the section about the phase modulator should say "The audio signal is given to the TR3 gate" - it should TR4 gate).
www.homemade-circuits.com
Amateur [ham] radio training and technical resources are a gold mine when you want to learn about radio design or RF in general; there are frequency bands dedicated for amateur use slotted in throughout the RF spectrum, from 1.8 MHz all the way up to the terahertz region.
(You still need an appropriate level of licence to transmit on any, and a higher level of licence to build and certify your own transmitting equipment, rather than using commercially built gear).