As I described in my previous question, I attempted a gradual conversion from a dérailleur to a Shimano Nexus 3 gear hub. I have installed the hub, keeping the dérailleur's pulley for now, with intention to remove it completely when I buy a new chain (this one is worn out anyway). However, I noticed when installed the wheel, the washer doesn't match my dropout, but I'm uncertain which one should I buy instead.
From your photo, it looks like you're fine. The load from the hub is being taken on the anti-rotation arm that you've attached to the lateral tubes of your mixte. So the anti-rotation washer (I'm presuming you're talking about the orange washer) is not being loaded.
Also, as described in the responses to your previous question, your dropout is just horizontal enough that you can get about 1-2 cm of movement - enough to tension your chain if you can shorten it, so that you don't need a derailleur.