Interesting question. It likely depends on how long they think it would take to build up the arm strength and fix his arm. If it's 15-20 days, then you go ahead and do it. If it's going to take 30-45 days then that's going to euchre the season and I'd love to see the Jays say f*** it - you're probably gone at the end of this year so you can DH from now on and work on your arm on the side. If you can get it back in shape, we'll play you at 3rd.
But - I suspect JD's pending free agency plays a factor in the decision. I can't see JD wanting to DH for an extended period this year at all.