So, check the voltage coming out of the stabilizer, with the circuit complete (gauge and sender connected). If it is less than 10 volts, then the gauge will probably read higher when the battery voltage comes up (alternator is working).
I still don't think that is the problem (at least not all of it); but if the VS isn't supplying 10 volts, then you can wait and see what happens after the battery issue is addressed.
PS, it's also possible the new VS has a problem. We talked about this a long time ago; inside the VS can is a standard linear voltage regulator IC. The manufacturer of the IC specifies that it needs external capacitance for stable operation, but the VS being supplied back then did not have the capacitance. The result is a product that works sometimes, and sometimes not.
I bumped into the same problem when rolling my own VS; my first attempt would work fine on the bench (using my bench power supply for a 12v source) but then not work when installed in the car. Output would be way low for no apparent reason. Adding the specified capacitor fixed it.
Moss' VS supplier may well have fixed that problem by now; I simply don't know. But AFAIK Moss never publicly acknowledged there was a problem.