Hi Stephen, nice to have you here. "Bout time too!
The LP Signature comparison to a (good) Fender P is nothing new to me. My Fender playing buddy here always says that about my LP Sig "good enough to be an old Fender". When you look at the woods, the pup position and the fact that Les Paul actually admired Leo Fender for forging a bass sound (he wanted to emulate it with a bit more flexibility), it's no big wonder. My LP Sig reminds me of my Fender Sting Signature in sound more than anything (and that is, yes I admit it, a great sound, silly Sting inlay or not), just a lttle blurrier with more billowing bass due to the hollow bodyconstruction no doubt. That LP Sig pup has singlecoil character in certain settings and adds humbucker grunt in others.
That is probably the core of my gripe about the Epi JC Sig: Mine just doesn't emulate that Fender sound with that presence shine around it like a halo, it rather thuds like you would in fact expect from a Gibson bass. It has nothing to do with the strings, it's just the nature of that bass.
Uwe