Actually .MTS is the container with AVCHD being the format used which, as I’ve attempted to explain a number of times uses MPEG-4 AVC /H.264 video codec and Dolby Digital AC-3 audio codec. Both of which are available in Shotcut for you to use.
I’m confused why you’re stuck with this to be honest.
I import, edit and export .MTS from my Panasonic HDCamcorder all the time and never have issues retaining visual IQ after editing and exporting. File size does not always equate to quality because it mostly equates to Codec compression ratios.
While not a totally accurate comparison, you could think of MTS (MPEG Transport Stream) as similar to DSLR still image RAW formats such as DNG (Digital Negative) which when edited and saved out to (commonly) JPEG are much smaller yet the image quality is either the same or due to good editing, is better. Yes RAW have more DR to work with while we edit, but what we save as JPEG is what we need. As too, MTS to MP4.
As an alternative suggestion, if you just want to adjust brightness then save out to a format ready for future editing rather than playback ( and retains a wealth of image information) then you could choose ProRes.