You've painted yourself into a corner. You've setup your network to be bridged, not routed (IPs in same subnet). Yet you have a Wifi client connection, which cannot be bridged to.
The Wifi specification prohibits bridging to an access point client connection. The access point will not allow it. That's why WDS must be configured in both access points. Neither supports bridging to an ordinary client.
It's time to rethink your setup. Here are four possible options:
Add a second access point and configure WDS in both access points. Connect wired machines to the second access point. This assumes your existing access point supports WDS.
Use a client station device. Connect this machine to that station device. That way, no bridging will be needed because each client connection will have only a single machine.
Add a second wireless router that can run aftermarket firmware like Tomato or OpenWRT. These have a feature called "client bridging" that uses a form of NAT to make this work (making more than one system appear to have the same hardware address). You can then connect any number of wired clients to this device. This won't require any changes to your existing access point.
Add a regular wireless interface to this machine and let it make its own client connection to the access point.
If you want to use NAT, you need to assign each end of the private link between the two machines a private IP address. For example, you can use 192.168.200.1/24 and 192.168.200.2/24. You will also need to make the private address of the machine doing NAT (the one with both connections) the other one's default gateway and you'll need to set its DNS server IPs manually.