You are already golfing buddies. You have already done drinks/dinner afterward. I don't see your problem, frankly.
The only thing you haven't done, sounds like, is ask him to accompany you to something independent of the golf. Doing that is tres simple. Try something like:
Hey, I just got tickets to the x symphony, or y game, or movie passes, or some show. Wanna go?
BTW - His comment about 'writing dinner off' is him trying to make you feel comfortable (so you don't guilt about him paying for you). Its a fuked up way to go about it, but nowadays you can't blame a guy at all for doing that. The 'rules' for dating are all messed up, to the point there ARE women who would take offense at him picking up the tab.
How is HE supposed to know you aren't that sort? Have you ever talked about it?
So. Just ask him to something already. Chances are he'll say yes, or have a genuine excuse to say no (don't freak out). I bet if he does say no, it will be with regret and he will suggest a different option.
Or, if he declines without followup, then that's your cue you have a golf/after golf drink buddy and that's it. Relax about it, either way.
Second thoughts can generally be amended with judicious action; injudicious actions can seldom be recovered with second thoughts.
--Cyteen by C.J.Cherryh