Yes, the outside diameter of a 125/90-15 is 23.9", this is an inch less than a normal 318ti wheel. So technically, if you're going to "harm the differential"
it would be when you run the 15" spare.
125/90-16's overall diameter is 24.9".
Stock 318ti sizes
205/60-15 overall diameter 24.7"
225/50-16 overall diameter 24.9"
Yes, I said he would need to get a new wheel, I even provided the part number. It is $35-$50 depending where you get it from.
I don't know if it will fit perfectly in the tire carrier. Potentially you could modify the stock carrier. You'd have to make sure there was 0.5" clearance to the tightest spot on either side of the wheel. Or just stick it inside the car for longer trips with a wheel cover on it.