Like Tinker,,I don't swedge the case head at all, just the brass ahead of it. If anything, sometimes I used to have to trim the head dia down on some 303 brass to allow chambering. Not so with the HXP stuff.
After the first firing it's neck size only.

My 220 Swift cases are .446/.447dia at the head.
From what I read,,and I realize I can't believe everything I read,,that's about right for the 220 brass.,,and about right for the M/S case.

If that's 'small' for the x53R it doesn't seem to be so (at least for my rifle). They form, load and fire just fine.
Some 1915 dated x53R Military ammo I have mics .448 right ahead of the rim

Makes fine M/S cases too if you need to.
No lop-sided swayback cases,,no cracks, just usable brass that go 6 to 8 loadings then I toss them.

No thick neck problem with the 303HXP,,just the small step at the base of the neck inside after forming. Probably heavier brass in the Military cases or just the thicker wall forming up where the new neck is being pushed back too.
The drill takes care of it nicely. ('G' drill, I went and looked. .261d and probably cuts a bit oversize so it's OK. Slips right into the case mouth and down to the 'step'. A quick couple of twists and it's clear.)

Keep it simple.