Which measurements can I take using which programs to exploit the already low quality and save as much space as possible?
There are a few video quality metrics available for you to use, primarily SSIM and also PSNR. You can use ffmpeg to convert the video and then compare the output.

Step 1 Convert the video

ffmpeg -i actioncamfile -c:v libx264 -crf 23 -c:a copy -map 0 compressed.mp4


The CRF value modulates the quality. Lower values produce better quality but larger file sizes. Typical values range from 18 to 28, but absolute range is 0 to 51, but 0 being lossless, which will produce an exact looking file but will be bloated.

If you want to test with only a small section, say 20 seconds of the source starting from 15 seconds into the file, you can use

ffmpeg -ss 15 -i actioncamfile -t 20 -c:v libx264 -crf 23 -c:a copy -map 0 compressed.mp4


Step 2 Measure the quality of the conversion

Run

ffmpeg -i compressed.mp4 -i actioncamfile -lavfi "ssim;[0:v][1:v]psnr" -f null -


This command output, at its end, will display lines like this

[Parsed_ssim_0 @ 000000000039ad80] SSIM Y:0.984483 U:0.980458 V:0.980921 All:0.983219 (17.751712)
[Parsed_psnr_1 @ 0000000000398320] PSNR y:42.63 u:43.19 v:44.09 average:42.90 min:42.07 max:46.16


You can aim for a SSIM All value to be at least 0.99, or set your own benchmark, based on what's acceptable.