I should have preceded my reply with a large caveat - I don't have one of the Swagger side-mouthpiece style HT's. Forgot about that. Mea culpa.
While it's conceivable the body could be submerged up to the mounthpiece joint, you'd have to leave the cap off the container and that seems too much a kluge. But I think it likely you could find a different container which permits submersion, with the HT either upright or on its side. E.g., we have a food container that is 6"w x 11"d x 12" h.
EDIT: The aforementioned
@t-dub method works well, too. Put water in the HT, place rubber stopper in bottom of HT, add small amount of PBW into top, plug/cover top, shake vigorously, rinse out (or flush if using his little plug/tubing tool). I like the submerge method better because I tend to be lazy about cleaning and just dropping the HT into the solution container overnight is very easy.