This would be referred to as a “no host” dinner. And, “join us” should be enough to let them know they are paying, as opposed to “be our guest at dinner” which would imply you are paying.
But, your friends might not know what “no host” means, nor pick up on subtle wording.
So, I would suggest something like one of these:
We would love it if you could join us for a no host dinner afterwards at XYZ restaurant. Dress is ___ and entrees start at X dollars.
We would love to have you join us for dinner aftewards. XYZ restaraunt offers casual dining and menu choices in many price ranges.
Join us for a Dutch treat dinner afterwards, at XYZ restaurant.