Via cost isn't a problem of PCB price. Most PCB manufacturers don't charge them separately these days, possibly except for low-cost high volume production. Costs in terms of occupied area, signal integrity and reliablity is a different thing. Marce has clearly metioned some points against a two layer design. I don't know the application details, so I don't see a purpose in adding guesses about possible problems with your design.
Two layer implies, that a continuous ground can't be achieved, except for simple cases that have effectively single side wiring. The next best solution would be a tightly meshed ground. If it can be achieved neither, you should
- go for four layer, or
- do it as two layer somehow, pray for correct design operation, keep away from any EMC tests and don't complain at Edaboard about possible failure