Current tenant is saying that nowhere in our agreement does it say he has to allow access to anyone for showings.
Do I have a right to request access (with 24 hrs notice) for showings?
What are you trying to do here? Are you trying to "win" an argument, or are you trying to minimize the rent money that you miss out on between tenants?
Your tenant is (not so tactfully) letting you know that they'd like to share in the motivation that's driving you. You could argue with the tenant, and they could sabotage your efforts by simply doing nothing. It'll also give your prospective tenants a taste of what type of landlord they might be dealing with.
Or you could figure that you're going to save a few hundred bucks by showing the place on the tenant's time, so maybe it's worth giving the tenant a couple hundred bucks. Heck, for $250 they might even pick up a little and say nice things about the place. You will certainly gain nothing by pointing fingers at the lease and bickering with them, and you're much more likely to lose even more.
If you're not willing to share the savings with the tenant, then you might end up having to wait until they move out... and under those circumstances I bet that you'd lose another day getting the place ready for showing.