The GPS module is an EB85A.
If it is labeled as a EB85A (and not FV-M8) then you need to check the firmware version using MTK's mini-GPS PC application. The firmware should be version M-core_1.85. I suspect you have an earlier version because I think I can see signs of the data lag issue that was solved by M-core_1.85.
TX is 500mw 900mhz from Range Video, ... GPS is relatively close to the TX, probably 7 inches or so.
At that distance the EB85A is expected to have inteference issues because the 900MHz/500mW Tx is a GPS signal killer (EMI/RFI report:
Click Me). If your model is small like the EasyStar, I'd recommend that you put the GPS module near the tail with the Tx at the front of the model. In addition, place a Toroid on the GPS host cable close to the module. Get 10+ tight turns of wire on the Toroid. Do the same Toroid trick on the TX's A/V cable, close to the Tx module.
with a -3db antenna that Vova included with the package.
If it is the wire whip monopole version then it needs to be upgraded to a good dipole. The stock wire whip causes stronger unwanted RF spurs at critical places.
Can the GPS cable be extended by soldering longer wires to it, or does it require some unobtainium wires?
Not a problem. Use small gauge wire so that you can get the most wraps on the Toroid cores.
Lastly, the unwanted harmonics are affected by the TX's RF channel. CH-1 seems to be less offensive than the others, but it really depends on the antenna that is installed and other factors. So, try all the RF channels and try to find the one that consistently gives the most GPS satellites.
One more question. When I place the clock on the screen, it is always off by 30 minutes.
Weird. If it is exactly 30 minutes off, then that would probably be OSD software related.