I've seen posts about ignition-related problems for the G1W, GS900, Mini 0801, Panorama II/S, etc.
One person with 2 Panorama cams had no problems in his VW, but both cams misbehaved in his Toyota.
In your case, you could try different firmware, e.g. If using v1.20 then try v1.17, v1.13, v0.59, etc.
If no luck, you could try a delay timer...
https://www.ebay.com/itm/360795585537
Alternatively, you could get into the habit of plugging the USB cable into the power socket adapter after each engine start. Inconvenient, but no more so then fastening the seat belt or turning the ignition key? Not forgetting that what doesn't bother one person can drive another to distraction!
You could try a different brand of camera, but there's no guarantee it will automatically start recording every time you turn the ignition key. The upcoming GuardTrak GT2S may be an exception, as I recall the spec includes an inbuilt variable delay timer. You may have to wait a couple of months for one, & I don't recall the expected price but, say, $200?