Took to mechanic and he is stumped.
Most mechanics know little or nothing about electricity. If it doesn't fit their list of "what to do when...", they are lost. So, don't be scared when a mechanic is stumped by an electrical problem. It's normal. No big deal. It just means it's not in his bag of "fixes".
Jay is right - with the GPs constantly on, your alternator hasn't got a chance. So, you know you have a problem with your GPS staying on, right? That's a separate problem from the charging problem - the GPs should not stay on, whether it's charging or not.
So, here's a huge tip any time you are troubleshooting a complex problem:
Fix known problems first.
Just do that. Don't try to figure out how it all fits together. You know one problem? Fix that.
THEN see where you are.
That's how you eat that elephant.
You know the GPs should not stay on, so concentrate on
that problem only. Use a trickle charger as needed to keep you going, but fix that GP problem.
Then you have cleared some smoke that should let you see the charging problem more clearly.
Might even fix the charging problem, but you won't know that until you fix the GP problem.