Thanks guys for all the suggestions.
Before using glue, I wanted to check how much the rope binding all through the handle would add to weight. I bought a cotton twine/rope from hardware store, tied it at bottom of handle, wrapped it all along the handle to the top, then tied it again at top. I added tape on top of the tied parts at bottom and top of handle to secure the tied parts. So far there was only 0.4 oz increase in weight. I then added tape all the way from bottom to top to make up for not using glue. So far 0.7 oz increase in weight. Added grip finally. Quite happy with end result of just 0.7 oz increase in weight and I could always remove the tape on top to have it at 0.4 oz increase. Bat feels very good now with the thick handle. Only thing though is that since I didnt use glue to bind, I am not sure how long it ll stay in place, but I decided to still use it the way it is now and see how long it lasts. Worst case if it comes off after just a few games, I will redo it with glue. All in all, very happy.