You really do need a directional antenna to track satellites and adjust for Doppler. I use an Arrow II antenna with built in duplexer(https://www.arrowantennas.com/arrowii/146-437.html) paired with a Yaesu FT530 for FM satellites, and the Arrow II antenna paired with 2 Yaesu FT817's for Linear satellites. Don't use more than 5W to transmit. I typically easily get in to the satellites using only 1 watt. As far as doppler, only adjust the UHF frequency during the pass for FM birds, and really only very minimally.
I do believe FTM300 should be able to work FM satellites in Full Duplex, so it's just a matter of programming the FTM300 with the correct uplink and downlink frequencies for the pass of the satellite adjusting for Doppler.
On this page is an invaluable cheat sheet for all the FM and Linear birds... https://ke0pbr.wordpress.com/2018/12/31/my-frequency-cheat-sheet/
Here is also a link to a spreadsheet that will show current and past radios capable of Full Duplex https://www.dropbox.com/sh/v3byggtuqw33fkk/AAAamatWbd9657AQfXhM6-wPa/Articles/Full-duplex_radios_for_satellites.xlsx?dl=1
A live satellite tracking app such as GoSatWatch for iOS or Heavens Above for Android will help you track the birds with ease.
Hope this info is useful! Satellites are very addicting and fun and are pretty reliable no matter what the propagation is.
Also watch these short videos on FM satellites