Reaching for the check in a group situation that doesn't have separate ones is pretty much a universal signal that the check-reacher intends to pay. I think that's what makes this situation so weird, and what makes your dad's initial comment totally fine.
In general, I think that if your aunt always initiates visits to your dad, the person who pays for lunch should be whoever suggests it and/or picks the restaurant. There are ways your aunt could visit with your dad that don't involve eating out, so if she's the one to suggest they get lunch, she should be expecting to pay for it (since splitting the check isn't usual). If your dad prefer to see your aunt over a meal in a restaurant and suggests that, then he should be expecting to pay for it.
Or, even though it can be a little awkward, your father and your aunt should just discuss when they make the plans who will be paying.