Is there a trick to mounting the new drive and its washers?

I can't see why you'd need to do it again.

But for trying to do anything similar in the future, consider using a _weak_ threadlock compound to hold the washers in place on the tray. You'd have to wait for it to dry. If course this isn't ideal if the washers _need_ to be able to move freely - you'd have to back the screws off slightly and 'fiddle' the washers loose with some suitable implement. It also is less than ideal if the screw in question is intended to carry any current. In that case conductive paint might work to keep the washers in place. If all else fails, consider grease, vaseline or even saliva. (But you'd have to consider possible long term effects of these - grease or vaseline would attract and hold dust, saliva could conceivably cause corrosion problems, although unlikely.)