Unfortunately, what you’re observing is a limitation of the MP3 format. MP3 encoders will often add padding to the start and/or end of files for a number of reasons, though the typical culprit is usually matching the audio to a set frame size. Even if FMOD decompresses the audio to PCM data, the padding will remain. If you’re interested in technical info on this padding, I would recommend taking a look at LAME’s technical FAQ.
I am unsure how exactly Reaper handles the reimported MP3 file such that it loops seamlessly, so I can’t comment on that, but the FSBank API and FMOD Studio’s bank building process work around this limitation by resampling the source audio in the last MP3 frame to make sure the audio aligns without padding.
That said, the best fix for this is to encode to a different format that does come with such a limitation, such as OGG which you’ve already tested.