Not that SSLstrip is very usefull these days.
But yes indeed, the client and the gateway (presuming there is any) will know each other because of the ARP table.
Both will do a broadcast saying who they are and will actively look who has IP address x.x.x.x.
The gateway for example will reply and say that IP address is at FF:FF:FF:FF:FF ,the gateway will then make an entry in its ARP table with the MAC address and IP address of the client.
Same goes the other way around.
So if you have that little bridge of yours (which works like a switch) you would need some way confusing the devices.
Basically lying about your IP address and have them send traffic to you instead.
Since you let this traffic through, neither side will be aware of any such thing going on.
Be aware that if you ARP poison the devices you have to make sure that you send out the correct MAC address for their table since you are using more than one interface.
Easiest way to do this is have 2 instances of ARPspoof running each for 1 interface.
Makes sense?