You add a tome of damage every time you upgrade the attack.
What part of that do you not understand... The tome gets used up and damage is permantly increased. No multiple items. In the case of the TDs, each kill simply adds one tome to the tower.
If the inventory is the problem, simply add the hero inventory, add the tome and then remove it. No one will see anything graphically and the unit will gain permant attack.
Also works for HP as well with causing no instabilities at all, but hero stats will cause a crash.