Have you checked with the office manager at the original vet you went to, to ask them if they will allow you to pay in installments?
Giving the cat away is not going to solve the issue, in that case she is better off being surrendered to the humane society so she will not be allowed to bring more unwanted kittens into the world.
A "spay & dash" as you put it also would not work as in most if not all clinics they will ask for payment before you get the cat so at that time they may also go over any instructions as to after surgery care with you, then they will retrieve the cat from the kennels.
Avoid biting when a simple growl will do
The Spirit Lives As Long As Someone Who Lives Remembers You - Navaho Saying
Vindication ~ For all those pets who became sick or lost their lives from tainted pet food