Regarding the '51:
It sounds to me like you fell in love with the look or idea of having a vintage style bass, but didn't fully understand what you were getting. You get a '51 reissue, you get what they had back then. As far as the bridge goes I'm going to assume that it's the two saddle design that you don't like. The modern P-basses all have 4 individual, adjustable saddles. Setting up a two saddle bridge is much trickier. Modern P-basses will also string through body (at least mine does). I'm not certain on the less expensive models, but the MIA standards do. A pickup change again will be easy, but you're not going to make the '51 sound like a modern P.
It's like you bought a fully restored model-T, and are complaining that it doesn't have an automatic transmission, power steering, and ABS. You are trying to make the '51 something that it isn't. Before you bastardize a perfectly good bass, I would suggest selling it to buy another modern P-bass, because that is what you keep describing. Again, there's a reason Leo Fender made all of those changes that are incorporated into the modern Standard Precision bass, and they are the things you keep comparing your '51 to. Sell the '51 to someone who will appreciate it for what it is, and just go buy an American standard. Problem solved. If I had that bass, I'd do absolutely nothing to it other than put flat wound strings on it, set it up, and play it.
Regarding the upright:
Opinions vary on the adjustable vs solid bridge, but it's logical to conclude that a solid piece of wood is going transmit the vibration better than one that's chopped in half. If it feels better, than that is more likely a function of the height of it being different than how you had your adjustable bridge set. There are many variables that affect the volume of an upright, but in general: higher string height = more volume.
An adjustable bridge is nice in four season climates because your bass is going to expand and contract with the seasons, and you can compensate for these changes by adjusting the bridge. Also, if you ever make a drastic string change - you're kind of fucked without an adjustable bridge. However if you are in love with your setup and string selection, then a fixed bridge will serve you well.