Michael, good suggestion. I did use a rubber jar lid gripper. But not in the position mentioned. I did try pushing "in" though.
Akira, I've just looked at the LF-4 and LF-1 each when placed on the helicoid. But that outer flange doesn't seem to interact with the helicoid. This is a factor to remember, however, when retrofitting various non-Nikon F-mount lenses or helicoids with lens caps. Make sure that outer flange doesn't get caught on anything.
While I like to consider myself "strong enough - for a girl", it has to be noted that I will never have the hand strength of the male of our species!! If my husband Michael had been home it is likely he could have made short work of removing that stuck lens cap for me. He plays piano and has very muscular hands & fingers, but was out in California judging the papers from a mathematics contest.
Eric: The problem you encountered is due only to the dimensions of your 'not-original' F-Mount.
I also think that is indeed the case. I do like the new style Nikon lens caps. They do stay put better as noted by Dave. But this 'not original' F-mount helicoid is going to wear an old style cap from now on.
Also note that even though the new cap might be softer plastic, it seems tough. It held up well under mallet blows and pry tools. I only managed to damage it with that not-original F-mount.