Makes sense. I would recommend against the UX90 (or similar camcorders). They generally don't too well in low light conditions, and good video is very light hungry, more than you might think. Also, if you have a high contrast situation, where some of your track lights create strong highlights on a person or instrument but you also have lots of shadow areas, you want to to look into cameras that support recording in log formats (s-log, v-log, etc) as that gives you more dynamic range to work with. That usually doesn't apply to camcorders like the UX90. Also the UX90 has a fixed zoom lens, which limits your long-term lens choices.
The UX90 has a 1" sensor, which is still tiny (it has a diagonal of 15mm, vs. M43 cameras like the GH5 has 21mm, regular video cameras are around 26mm, and a full frame DSLR like Canon or Nikon are 43mm). The bigger the sensor the more light it picks up and also the more depth of field you get.
Given your budget and that you likely want to keep lighting similar to what you have now, I would look in the DSLR style range and look for cameras (and lenses) that are good in low light. The GH5 supports v-log. New Canon cameras now support c-log I believe. And some of these cameras support 'dual native ISO', which means when you film at higher ISO (due to low light), the noise will be more manageable.
That still doesn't address your synchronization entirely. You probably will end up having to do it via waveform match, as the camera choices that support timecode solutions probably limit you too much at this budget level.
Resolve is a good software option for, it's free up to UHD, and just $299 for the full 4K version.
Last edited by allklier; 21st January 2019 at 02:20 AM..
Reason: extra detail...