Spray mist coolant would be a very good move. Machining without some form of coolant is a pretty big handicap - both to working speed and to tooling life. The largest cut I ever took on a lathe with carbide was about ¾" per side at .077" feed rate. That was a 52" swing lathe, an old German made machine called Schiess. I served my apprenticeship in a very large shop that did a lot of repair and new work for the big steel mills - US Steel, Inland, I/N Tek, Arcelor Mittal, etc. The largest lathe was a Niles with 72" swing, it had something like a 200 HP DC drive. That was a cool place to work.
Water soluble coolants won't cause rusting of the lathe unless they're mixed too thinly. Mixed in the correct ratio, they leave behind an oily residue - at least the good quality ones do.