I wouldn't "strive" for raising the volts as high as possible. Something like 13.5 is where I would strive for if it was me. You want the cells around 3.35 or a bit higher from my understanding. Your not trying to charge the battery your trying to bring up the weak cells so they can catch up with the full ones.OK, that sounds pretty much like what I'm doing right now with my desktop charger. I can set it to exactly whatever voltage I want and tinker as needed.
And I've already been doing the "fun part", trying to figure out how high I can set the voltage without having it trip into over-voltage protection. Right now I think I'm at 13.86 volts or so. Going to go out to the shed later tonight and check on it.
Your overvolt problems are from some of the cells being full not the entire battery being full.