Good luck on your trade, it sounds reasonable to me. They're both midrange instruments and if I'd have bought the P-Bass and not been enamoured with it, I'd consider the T-bird in trade.
I was lucky enough to play the T-bird a few times before commiting to buying one (at one time I was going to order it, online, not played). Ironically, I liked the T-bird's sound and feel when sitting on a stool playing it, but as soon as I stood up and put a strap on it, it just didn't hang right for me. So I can see how you say you're not 'communicating' well with it.
The Fender Jazz Bass I'm going to buy "sits" well when on a stool or strapped up and standing, so like you, I'm off the T-bird bandwagon, despite them being good basses.
Again, Good luck in your trade search, someone will come through.
Rock on,
'Bolt