I hate to say it, but I think you're out of luck. <CUSTOM0> is the correct token and it gets set upon an influence increase or decrease. Try calling an influence script, and then call the token again in a new conversation. The problem is this function is appears to be hard coded in the influence script functions.
It's possible that it's getting the name from somewhere else, though. But I know it's not from npc.2da or influence.2da. It could be some other TLK entry; I searched for a few party members and I always saw more than one entry with just the name. So uh... try changing them all? That's really all I can think of.
As far as using tokens for a new influence system... well, that's not what's going on here. The token is used to display the name of the character whose influence has been adjusted, that's all. You could make a new influence system, theoretically, but I don't think you can do it with tokens; I believe they get reset upon loading a module. That's what all the <UNRECOGNIZED TOKEN> business is about. You'd need some new global variables for that sort of thing.