I'm the one who originally asked the question via PM.
All M42 use the same oil pump. The M44 uses a larger capacity oil pump.
Cross referencing the early M42, late M42, and M44, there are differences between them all:
- early M42 and late M42 share same cam sprockets (1121124733811311721887); M44 uses different ones (11361247951)
- early M42 and late M42 share same slide guide between the sprockets (11311721641); M44 uses a different one (11311743046)
- late M42 and M44 (11311743262) have a different chain tensioner guide than the early M42 (11311727342)
- late M42 and M44 (11311734694) have a different chain guide below the crank sprocket than the early M42 (11311721419)
- late M42 and M44 (11311739130) utilize a guide in place of the deflection wheel/gear found on the early M42 (11311727569)
- M44 uses higher capacity oil pump (it's thicker) than early and late M42
- the M44 uses a different crank position sensor than the early and late M42. The mount location is also different. Looking at pictures, M42 chain cases have a mount for the M42 sensor, however the M44 don't. Does anyone with a very early M44 have this mount?
Given I have to replace the chain case, I'm going to get a late M42 style at a minimum to get rid of the deflection wheel. However, if it's not too much hassle, I would love to use the M44 and get the larger oil pump.
Chain case part numbers:
early M42: 11141727191
late M42: 11141739699
M44: 11141739800
Here is a picture of the mount (where 11 fits into 1):

What can I do to use the M44 chain case?