The Twins have used 25 different pitchers this year, not counting Nick Gordon, barely into June.
I think bringing up "minor leaguer relief pitchers" to see what might stick has been tried.
Indeed, I'd say building your bullpen plan based on waiver wire pickups and minor league flotsam and jetsom is how you get into this position in the first place.
Find external major league help, which won't be cheap, and put your best minor league arms (currently starters) in the pen.
And yes...don't trade away your best relief option(s).
Cycling through what's remaining in St Paul's bullpen accomplishes nothing. Shuffling deck chairs on the titanic.