I had days of frustration, same kind of yours, just to conclude that I made my same old mistake...
You can see in this thread, how the Moitessier is sensitive to the quality of the power source:
https://forum.openmarine.net/showthread.php?tid=1368
It was mainly about AIS reception but I can imagine that will be true for GPS as well.
While trying to troubleshoot the "hacked" moitessier configuration on a desk, I used my smartphone charger to power the RPi, the GPS reception was appearing sometimes than disappearing for good, at least I had a confirmation that it should work overall..
Finally I ended up connecting my beuatiful industrial grade power supply that I had to disassemble from the target device and bring to the desk, It works as a charm now.
TRACO TMDC 20-2411
I love that power supply for several reasons - low noise and stable, has protection so I am less afraid to burn down my boat, wide range of input DC, so I can use my device both with 12 and 24 V DC, but also take any old laptop charger from scrap and use it for 230VAC operation at home.
The GNSS status LED was blinking normal in both situations ( at effective satellite reception and uneffective), so maybe actually the communication itself was adversely affected by bad quality power supply? I am not sure if the green LED blinks normal if it gets fixes..?
(please mind that I am a mechanical engineer, know nothing about RF electronics)
I wish you good luck with that, after all, lovely piece of abandoned equipment