I've had several MGB's over the years. Enjoyed them all, to varying degrees.
First of all congratulations - she looks lovely, and also well done for buying the chrome bumper - one of my cars was a rubber bumper BGT and apart from the spoiled looks the ride was not as good. Something to do with different suspension set-up and the car being slightly raised.
Mechanical problems tend to be few and far between with the B and Midget. The parts are all standard, and well-tested. I would be looking at fuel-pump electrical connections if I were you, and of course it's possible you have a faulty pump. Your little plastic fuel-filter should be mostly full. Rot was always the problem with these cars, but presumably that's all been taken car of since you bought from a dealer. Oh, and stick one of those maintenance-free high-amp glass mat batteries in. You probably used one on the bike. They are great! Once fettled you shouldn't have to spend much on maintaining the B.
You've bought at the right time of year since you have the cold and wet months to do your troubleshooting before next Spring.
I had exactly the same problem with my three-wheeler, and broke down a few times last Autumn. We ended up changing coils, plug leads, the fuel pump, battery, stripping the carbs and refurbishing them, fitting a new electronic ignition system, and having a professional wheel balance, - this year since June has been trouble-free and the Triking has been a joy to drive.
ps - you evidently didn't miss anything with the pre-war Morgan. Lovely - for someone else to own.