We have these crazy speed drills too. They started with addition and subtraction and then, obviously, mult. and divis. they were a source of anxiety for DC in the beginning but now, they are second nature. We practiced them at home and I gave DC tips like, don't focus on neatness, but try to be legible, do NOT erase, that adds seconds to the time, cross out and rewrite next to it, and don't say the problem in your head (ex. DC was saying 7X8 then writing the answer) I said to just look and write the answer because all of those little things added up and tacked on time. Those tips were all it took and we are a pro at it now. I don't think there is anyway around doing them. DC started them in 2nd and is still doing them in 4th. (though we skipped 3rd, they obviously still do them in 3rd if they have to do them in 4th too.) That is just a small portion of math and though I don't agree that speed is a necessity (to that extreme) I think it's just something that takes 1- 2 minutes of their math time and I can overlook.
Perhaps starting small (try 10 problems in X time and make short term goals and work your way up to the full amt.) Sounds like there isn't a way around it. Make personal goals for him instead (if you know he can normally only do X, shoot for one more than that. Regardless if he completes to the teacher's standards, he met his personal goal of improvement.) That might increase motivation.