Perhaps the confusion is in the title die to the word 'sniffing'. Apparently that's what you want to do in the end but it is not the subject of your question.
My understanding is that the bridge itself is doing what you want already, i.e. on a router with 4 + 1 ethernet connections and the standard lan (4) + wan(1) setup all traffic passing the bridge is visible on all four outputs.
So if you connect another router to that lan<>lan and with as default wan gateway of that second router wherever you do want to do your sniffing you can watch the traffic as you like. Make sure using iptables that whatever return traffic would be generated by this 'parallel' machine gets dropped so it seemingly isn't there.
Have never tried this, so it's just a line of thought and my nderstanding of what the bridge does may be wrong.