Pretty funny. I have a very similar situation.
I picked up a used Squire Jazz (Red, like your P, which caught my attention) and want to upgrade it a bit too. I bought a used Hipshot B bridge, used fender style tuners (since the Squire Jazz tuners are terrible looking and lose their tuning quickly) and used EMG pickups to drop in it. I also got a new pearl pickguard to put on it as well. The white is a little bright for me. Might be $120 in parts, with new strings. The classifieds here on TB rock.
The bridge might require drilling, which I am nervous about, but hey, it's not a real Fender, or a $600+ bass. I figured WTH and will give it a shot. The bridge will cover any extra holes anyway. Even thought about making it a string-thru body but didn't commit to that yet. Maybe on my other cheap Samick just for fun.
I'm going to build this up and try it out. Hopefully will drastically improve the sound. Like you said, you liked the feel. I felt the same way when I picked this one up. Take a cheap bass and fix it up? I'm in. Besides, practice makes perfect, right?

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60th Anniv Fender P, SX J, Epi T-Bird, FrankenFretless 4
Hartke Club Member #68, SX Club Member
Bassist for:
Crazy White Dog