Usually with a Carter you will run a pressure regulator, though there are people here that aren't and I don't get how they don't have that same issue.

My general assumption is that the factory pump may just be worn enough to lower the output pressure to something the Weber is fine with.

I know everything I've read about Weber carbs as they relate to line pressure is: They are very very picky and even a tiny bit too much pressure will cause issues.

I have a 38/38 and it was suggested perhaps I get a fuel cutoff solenoid off of EBay as mine likes to do the same. I believe it replaces the main jet.