I'm not to sure that any rifle could develop a 777 "crud ring" with a single shot. As stated, some rifles don't seem to develop a crud ring at all. IMO what you are experiencing is just a tight fitting sabot/bullet in a barrel that has normal fouling with 777 use. If you used a looser sabot or something like a Powerbelt, which loads easier, you would be able to shoot a few shots before cleaning.
Either way, it is a good idea to run a damp patch, followed by a dry patch, down the bore between shots. When afield you shouldn't need to do this for a follow-up shot. Typically, looser fitting projectiles are faster and easier to load but are not quite as accurate as the tight fitting sabots.
Reportedly the new Buckhorn powder by Winchester will allow 40+ shots without any cleaning at all. Let's hope the reports are true.
Just my 2 cents