I would transition tag him and check the wind before FA and the draft. If I got the feeling he was going to be a no show, I get a FA QB, draft a QB in the 1st, possibly trade up for Tua and rescind the tag after the draft. Tua is a lefty, so was Moore, I like that.
I tell my new HC that the plan has changed and he gets to bring Tua along slowly, do what he can with the FA QB and I immediately add a year to his contract giving him a 6 year deal. The circumstances changed after his hiring and I'd want him to feel good about the change. In fact, he might be more in favor of the change than the GM.
And think of the merch possibilities for Booger. On opening day 2021, Tua can put lei's on all the women entering the stadium and Booger can sell t-shirts that read "I got Leid by Tua". The concessions can sell Cowpoi. And best of all, Rowdy can do a torch dance at halftime and maybe set himself on fire.