Hello matey, just spotted that you're having problems with the starter circuit and, correct me if I'm wrong, but it appears that the starter, the solenoid and the battery are all satisfactory when off the car. I wouldn't like to profess to expertise here, but there are a few other things that come immediately to mind. The first is the more obvious of the two, which is to suggest that there is a fault in the wiring or switch from the battery end through the ignition switch to the solenoid, though it strikes me that my second thought on the subject might be more likely; during your rebuild, were you quite liberal with the paintbrush? Especially around earth points? And I refer particularly here to whether you painted the starter, or the block, though any earth point that's been painted might not have full integrity, no matter how tight the connections are. I had this some time ago, and only found the fault when I took a direct load from the live side of the battery direct to the starter with a high-current cable. If there's anybody reading who isn't absolutely au-fait with motor electrics, leave it to an expert, but it sounds to me like an earth that hasn't fully 'made', which would also explain the slowly-climbing meter as the current tries to flow. It might also be an idea to feel carefully about for warm connections as, if this is the case, the resistance presented by the poor connection may well generate heat at the point where the fault lies. It's also possible that there's a fault in the reg box (or, indeed, a poor connection here, too) if you still have a dynamo on the car, or in the alternator regulator if the car has this fitment. Failing all else,....bigger hammer.
I hope that this is helpful to you.