I have not used Bluetooth speakers with in-built FM Radio so can't say much about them however if you want a radio with solid FM reception (you'll get MW and SW reception as well) then search for Xhdata on Amazon. The Xhdata D-109 and D-808 are good radios. I personally use a Tecsun PL-380 but Tecsun seems to have almost disappeared from the Indian market. The Xhdata D-808 is almost like a Tecsun PL-380... do check some written reviews online and on Youtube about the D-808. The D-109 is 3.6k and the D-808 is 7.4k on Amazon at the time of writing this post.
I have the above Tuner as well but the tuner will need additional speakers to connect to for audio output and also I feel the Xhdata will have better and stronger reception across the bands (based on my opinion using a Tecsun PL-380) and you'll also get a portable radio with MW and SW as bonus which actually work.