Or would it screw up the flavour?
The dill in the tzatziki might throw off the flavor in the spice paste.
I don't think that would be a good flavor fit. I'd either get to the store or wait to make this dish another day when you have what you need.
Canned coconut milk might be a good substitute if you don't have any dairy.
honestly, the flavors and spices from the tikka paste will overpower the dill. Plus the tanginess of the tzatziki will probably really help the sauce.