I have some suggestions to get you going, but first we need to establish how much you really want to keep the MTS format. There is no speed, quality, or file size advantage that comes with exporting to the same format as your input files. Recompression is recompression from scratch, every time. Nothing can be salvaged from the input file to reduce processing time or preserve quality, and even if it could, it would be made obsolete after applying any filters such as brightness. In fact, you could gain quality and file size improvements by using a newer, smarter compression format.
Assuming you’re willing to use H.264 without AVCHD/MTS restrictions, here are some options. First, on the Export tab, choose the “H.264 High Profile” preset under the Stock header. Then click the Advanced button and consider the following changes:
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If you want the exported file size to be guaranteed identical to the original, go to the Codec tab and change “Rate control” to “Average Bitrate” or “Constant Bitrate” (preferably Average for better quality). If you know the codec bitrate of your camcorder, enter it directly and you’re done. If you don’t know it and can’t find it in an owner’s manual online, you’re down to some math to reverse-engineer the bitrate… (File size of an input MTS video in bytes) * 8 bits per byte / (Length of video in seconds) - 128,000 bits per second for audio = bits per second for video. Divide by 1,000,000 to get megabits. It is often possible to produce higher-quality video by turning on the “Dual pass” check box, but that requires processing time you may not wish to spend.
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If you can flex a little on the output file size (we are still on the Export > Video tab here), you can often get noticeably better quality video by using “Quality-based VBR” rate control. The default quality setting of 60% is quite high and makes a great-looking albeit large file. It correlates to H.264 CRF 20. Finished videos in AVCHD are more like CRF 23 to 28, which in Shotcut is between 46 and 54 percent on the Quality slider. Try a few test exports in that range until you find a file size and quality trade-off that you like.
Now to address processing speed. This applies regardless of which option you chose above. Go to the Export > Other tab and change medium to veryfast on the vpre and preset lines. Considering the nature of your input files, the quality difference should be unnoticeable, but encoding time could be faster than real-time depending on your CPU.
The result of all this will be a .MP4 file rather than .MTS. Could you let us know the results of those settings and we can investigate from there?