So I ask you, how weird is this? Any sexual activity with an animal that is invited or facilitated by a human is bestiality.
I believe things have to flow both ways for open or poly relationships to work—every party must be at peace with not just having sex outside the relationship, but with their partner doing so as well.
Without launching your own one-man internal investigation, if you want to keep your conscience clear, avoid dodgy establishments that offer services significantly below market level, have blacked-out windows, or require a buzzer for entry.
Can the act be consensual when one party has the economic power? This is a stimulating thought exercise, but there are a few issues casually raised in your question that I think are even more germane to your immediate reality.
Start by … talking to your wife.
Getting eaten out by a dog is a choice.