|Bridging two network cards is not the same as what you are trying to do a totally different thing altogether.|
The IP of the router is the gateway address. Every device connected to the router will have that address. It is not the IP address assigned to card or wireless adapter which is the one that matters.
Yes the router will send the data to the assigned IP for incoming traffic, the same IP that initiated the transfer by sending in a request in the first place. The router cannot send incoming traffic to an IP different than the one that initiated the transfer in the first place.
It is the routers software that assigns the IP address to send incoming data to based on the IP address of the device that requested the data.in the first place.
Even so, splitting send and receive in as you ware suggesting for Internet traffic, if it were possible, would be a pointless exercise as the bandwidth of the slowest LAN connection is a lot greater than any Internet connection. Nothing would be achieved.