How to measure EMI without special equipment?
Look at "SDR dongles" - USB dongles made ostensibly for laptop reception of TV broadcasts, but repurposed for amateur radio purposes and sold for tens of dollars.
Not all of them cover the spectrum up to GPS frequencies but some do, so shop around.
Combine "SDR dongle" with "spectrum analyser" and "open source" search terms and I think you'll quickly have an outline of a solution - not a precision tool but something capable of diagnosing EMC problems and, if necessary, pre-qualifying equipment before expensive compliance testing.
Without doing any research I would suggest it's at least five orders of magnitude more sensitive (detecting microvolt signals) than your schematic of a diode-based wavemeter (detecting 0.x volt signals), so it should show most local interference even if it can't display a GPS carrier as a peak on the spectrum analyser.
A couple of minutes on Google suggest that they ARE sensitive enough to detect GPS signals presumably through the usual GPS techniques for extracting information from the noise floor. This doesn't prove much in itself but gives me some confidence that this is a viable approach.
If anyone has more specific and detailed recommendations, please add a better answer (if it isn't too close to a shopping Q&A).
The best receiver you have is your GPS (or a friend's). Without locking to the spread spectrum code to reduce the effective measurement bandwidth, the signal level is waaay below the ambient thermal noise.
What facilities does your GPS have for showing the strength and quality of the satellite signals? My car Tomtom only tells me how many satellites it has acquired, when it's locked to them. My walking Garmin OTOH has a page that gives me strength and quality of signals for all satellites it can see, whether they are fully locked or not.
On some open ground, run a suitable GPS receiver, look at the satellite quality, and then bring your dashcam up to 3m, 2m, 1m, you get the idea. This will enable you to get a qualitative measurement of whether it is indeed the dashcam, and then how well anything you do to it works.
It might be that energy is being emitted in all direction through the plastic case, in which case you're lost. It might be that most of it is conducted down the supply cable, and a suitably sized ferrite 'stopper' on the cable will improve things no end.