Hi Stan, if you are experienced in the development of PCBs with antennas, then you can try a two layer PCB. I have practised this several times, but in my PCBs the ground islands on the two layers are well connected. To reduce the cost again, all components are on the top of the PCB. On the lower side of the PCB is only the holder for the three AA cells.
Do you have the experience? Can you put all components on the upper layer? Do you create ground islands which you can then connect together? Your answer is yes - then measure the result with a Vector Network Analyser. If the result is good, then take your spectrum analyser and measure the emitted interference radiation. Then you take the measuring system to measure the radiated power in three axes and check how good the radiation is.
If your answers are no, then you better continue with four layers. You can discuss the details and the review of a PCB with your antenna consultant.
You are also welcome to send me an e-mail as a consultant on antennas: Harald.Naumann (AT) LTE-Modem.com