It's sort of a different variation of the "Tigers should trade for Frankie Montas" thing that comes up occasionally around here....
The pitching, even with all the injuries, isn't what is responsible for this record. And for all the injuries to the rotation, it's not like they are wearing out the bullpen... they've had one scheduled BP day so far this year and you can go days at a time without seeing any of the guys in the pen at the moment, except for maybe Lange or Chafin.
Just ride with what you have, plug Alexander, Pineda, etc. in when they come back. It isn't that hard.