I think you better have a family-member brought onto the field. So this one can keep the scorebook, fair and straight. Better have the league appoint scorekeepers...
It sounds like you're talking about kids playing in a competition. Parents, coaches and all the other adults involved in minor-games
shall always give the good, honest example!
As the umpire you have to rely on that. I would, in this case, call both coaches together and explain that Homebook can't be trusted. Therfore home-scorer is to be thanked (for nothing) and be excused (= leaving dug-out!!!). Scorebook that can be trusted now is official... Play on.
Home coach is having problems with that? Call it a game; if a team won't play fair, we aren't playing at all!
Make sure you collect all the needed papers, i.e. scorebook sheet from both teams. No one can complain about you at the league director. You did what had to be done. Show the books at the director, he will forfait the visiting team...