.366 bore?
That would make a groove of roughly .374" or so, you think?
The heeled bullets I have are .380" which must be for a .380 Rook or pistol cartridge.

Occasionally you do see rook rifles which are engraved for "380 or 360No.5 Cartridge" and shoot both interchangeably.
I have pushed these bullets through a .366" Lee sizer thinking the heel would be compromised but they do still maintain a heel lip to seat against.
My rifle has a bore of .362" so that is why I chose to load with black powder for the first go around. My patched bullets may work in your rifle. I have some 190grs ready if you want to try something that heavy in the faster twist rifle. The 1/50 may need something about 100grs, not sure.
I will post some pictures of the swaging process if there is interest in seeing that.