There are plenty of wired remotes. However, the camera has something like a 1 second delay while it finishes up closing the previous file, adding the metadata, and opening the new file.
I have come up with a solution that you may want to adapt.
I wanted to mark highlights in the footage as it happened so that I could find it in editing and not have to watch the full gains again.
Version 1.
I put a small microphone plugged into the second xlr record input witch the sensitivity set to not very sensitive. Then I put the microphone close to my mouth and would say oooh into it for a highlight. This recorded on the second channel of the audio. In post, I could quickly find the tones and place my cuts.
Version 2.
Then I had a small tone device with a button placed near the microphone. When a highlight occurred, I pressed the tone device which recorded a tone on the second channel of the video instead of people near me thinking I was going goofy.
Version 3. I bought a different tone device and electrically connected it into my microphone jack. ( I am an electrical engineer ) This eliminated any background sounds and was easier to find on the timeline.
One of these might work for your application.
John Lenihan
LeniCam Video Productions
https://www.lenicamvideoproductions.com