A few thoughts and suggestions as a result of some 20+ years shooting a PH P61 Artillary Carbine (AKA musketoon).
First of all the mould I use is the
Lyman 575213PH which is the mould that Parker Hale had made for their rifles. I have cast and fired several thousand bullets I have not had one tumble. Neither have I had to resort to any kind of sizing die.
In all cases of tumbling I have observed it was because the bullets were not up setting into the rifling as a result of either not enough powder or the bullets being cast out of something other than pure lead. As you have tried upping the charge it looks like your problem is the latter.
Standard load I use is 65 grains, by volume, of fine; either Henry Kranks or FO Triangle. Only difference is that FO Triangle is slightly cleaner. With either powder I can get off a full card without having to brush out. Lube is 50/50 (by volume) Beeswax / Vaseline. On the official MLA target I can usually keep all my shots inside the "8" ring at 100 yards.
The rifling on the "musketoon" is not shallow at all but progressive depth. That is it is deeper at the breech end than the muzzle. A variation on the "squeeze bore" concept. Just as it was on the original. One interesting consequence of this was that the variation that Parker Hale supplied smooth-bore for going on shotgun certificate were smooth bore at the muzzle but rifling gradually appeared as the further down the barrel you went. Lyman made an oversize version of their minie mould that apparently worked well in these barrels.