Can I test my directional coupler using a 2-port VNA instead of using a 4-port VNA?

Yes, a 2-port VNA like the Bird BNA100 can be used for directional coupler testing. The key will be to ensure you apply a 50 Ω termination to the unused port. For instance, in the setup where you would be checking coupling power, you would connect VNA ports 1 and 2 to the device's input port 1 and coupling port 3, respectively, and apply the terminator to the remaining (output) port of the coupler as shown in the image below. 

This is equivalent to using three ports of the BNA1000 where coupler's output port is terminated by the VNA port it is connected to by virtue of internal switching in the VNA.