I've checked in a car seat on LGW-FAO a few months ago and it was entirely smooth. Bought a cheap seat (for an older child) and checked it in in its box, which was not vastly bigger than a wheelie carry on. At LGW F check in they tagged it and told me to take it to the oversized baggage drop off in the middle (didn't make sense as wasn't that big).
The box arrived on the oversize luggage shelf at the back of the FAO baggage hall about 15 mins after the bags started arriving on the belt. No damage and equally smooth on the return.

I think a box and padding/packing should be fine. I'd invest in some proper packing tape as opposed to Sellotape, it's much stronger and is available in a clear colour.

Hope this helps