Hey,
First things first have your immobiliser removed
I too have been stranded by a faulty immobiliser and I had it removed on my 1993 WRX (aftermarket).
There are many things which could be causing your issue. But if your battery is strong, and alternator working correctly I doubt they would be related. Make sure there is no water or condensation in your fuse boxes. Try testing the battery first, weak batteries can cause weird issues!