Can't one of the valet parking outfits (Purple Parking or the like) collect the car from T3 and drop it off at T5 ?? .. shouldn't be rocket science as they serve both and I'd argue for no change in price from the £100 ish for 2 weeks. Give 'em a ring.
Appreciate the potential issues with off site valet car parking but used to check the crumb trail; on the GPS in my old Jeep and it was inevitably taken the shortest route from T3 short stay to a compound a couple of miles away so got a bit more relaxed about it. Unfortunately the Audi's GPS doesn't lay a crumb trail.