One problem with the bullpen structure is that there are a lot of guys who cannot be optioned out when they are overused. It's kind of fundamental to bullpen structure under the current rules to have a lot of guys who can be shuffled on and off the 26 man who can throw hard and pitch in the 5th, 6th, and 7th if needed. I think the only optionable guys in the bullpen are Finnegan and Garcia.
Our minor league 40 man roster bullpen arms are Adon (now), Willingham, Brzykcy, and Cole Henry. Henry is still rehabbing. Brzykcy has thrown 15 innings over 3 levels (4 in the florida complex league, 10 in A+, 1 in AA) and seems to be a bit walky. He's going every 3 days or so (once with a day off), so I'm thinking he's still not quite ready for a call up. That said, there's no reason Willingham is on the roster if he's not available for a call up.
Replacing Weems on the 40 man with somebody like Cate would start to create space to give prevent bullpen burnout. Maybe the move is to replace replace Weems on th 40 man with Cate, then call up Cate and Adon to replace Weems and Garcia for now. Hope Brzykcy is usable soon, and be prepared to shuttle in Willingham for Adon when he is burned out. Otherwise, can Willingham if you won't use him.