I've done DIY diamond coring of cement. Go slow and be careful. Lava rock is way softer. A 3" bit that can go 6" deep is around $75 on amazon (https://www.amazon.com/gp/product/B017JO7OVW). You can get an adapter to run it on a normal 1/2" chuck drill. I did this and choose a cordless drill I owned with variable torque settings - set it low - it will slip if you start binding, but that will save your arse. Don't use a strong drill that will bind up and toss you around. I built a guide out of plywood (using a jig saw) to help line up the hole and keep the bit from wandering - and had a helper spray water on the bit continously with a spray bottle. Worked like a charm.... You could build a holder out of plywood for the rock with a guide hole on one side.