Well that is odd that it doesn't happen with Federal or CCI. CCI used to be the hardest primer but not any more (if you are using newer primers). But since you have the full power main spring, that is odd since it should set off ANY primer, unless it is old and worn and lost it's strength. I won't insult your intelligence by asking if those are Large RIFLE Winchester primers.
I guess less likely things would be the hammer rubbing on the side its slot in the frame (you would see scratches on the side of the hammer), but that usually only affects guns with lighter springs. Maybe check to see if the firing pin is worn and not protruding through the frame far enough.
Maybe somebody else will chime in. I don't really have shooting experience with Remingtons, but these things are pretty universal. You might consider posting this in the gunsmithing section. Some people only look at specific forums.
Light hits are a pain. Good luck!