What is the best finish for a Ballard cast reciever, will it hot blue, color case harden,or rust blue?

IMHO,
The original factory finish was probably case hardened in colors.
That being done, will dictate to some degree how you refinish it.
Hot bluing is probably not a good idea; as it may turn out a weird "coppery" color.
This was true of Stevens 44 cast actions as well. I've seen 'em ! Check out G.B. item # 363656857
As previously stated, rust bluing or re-casing would be good, albeit expensive options.
One thing to be cautious of: I have read/heard that re-casing can sometimes shrink and/or warp an action if not properly done.
Another alternative that can look pretty classy would be to nickel plate the frame; then rust blue or case the component parts.
A satin nickel might display better than bright; but that is in the eyes of the beholder.
Good luck with your project !