Lava rocks are very rough, it's a good thing you removed them. Mine once cut himself against some lava rock that I used to have, he had a visible gash across his body but fortunately it healed. I don't recommend it as decor, especially since bettas seem to enjoy squeezing themselves into tiny spaces and lying on various surfaces. And if they try to rub themselves against it (i.e. flashing) it can definitely cause damage.
Sometimes, a betta's fins will tear on their own due to e.g. excesive flaring, but more often than not it's related to sharp objects in the tank. Try taking anything rough/sharp out of the tank and see. If you don't want to remove the hide then maybe try turning it in such a way as to make the sharp bits inaccessible to him (if you can?). Or try sanding it down somehow? Try the pantyhose test - run some pantyhose against it and if it snags, it's too sharp for the betta.
For the time being, the most important thing to do is keep the water clean so that the damaged part heals and doesn't get infected. It should grow back very quickly.
(on a side note, bettas with heavy, over-the-top finnage like yours are much more at risk of developing health problems, fin rot, collapsed fins, tail biting, as well as other issues related to inbreeding etc. Carrying all that weight around is very uncomfortable for them.)