There may be dirt/carbon under the seat. First, this may make it come loose again. Second, obviously the seat is not guaranteed to be square to the bore now if there is dirt under it, so when you try to lap the valve you may find you can't get continuous contact all the way around.
If the engine hadn't been run with the seat loose, the odds would be pretty good. However it has obviously been loose from some time (the spitting on shut-down that went on for a while before it quit working completely) so I think there will be garbage under that seat. It might work anyway, though - time will tell.
