Could we use the TPE to get Gortat, and package it with a couple of draft picks? Do a sign and trade??
Yes, this is my preferred option.
A bonus side that might help too, is that without Gortat, their PF talent and depth takes a big hit. And more importantly, by getting Gortat off their hands they themselves should be under enough of the salary cap (if I did my math right) to absorb someone like Brandon Bass themselves, a player that they should want on their team in my opinion.
So the idea here is convincing Washington to play ball on a trade, and I think getting Bass for their troubles, which is an expiring deal, should be a good enough incentive.
Heck, we could trade Bass for Gortat and not use the trade exception... it'll all depend on Gortat's starting salary.