Only one real reason not to do it: Money.
If they break up and the one who keeps the kid needs money, guess what? She's going to go through State Welfare and they will require that they tell who the father is. Then, the state will come after YOU for back child support and future child support until the kid is 18. In fact, if the kid decides to go to college, you could be on the hook until the kid is 24. Bear in mind that the state will do this NO MATTER if the custodial parent does not want to inflict this on you. She will not have a choice about it.
Even if the state doesn't go after you, it's very possible that the custodial parent could come after you and get child support regardless of how friendly she is now. People desperate for money can begin to change their views very quickly.
You MAY have a leg to stand on in the case of medically assisted sperm donation and an in-vitro fertilization but I really wouldn't place any bets on it.
This is an issue which has not been decided by the courts yet. You're playing with fire and with your family's future financial security.
I assure you, it won't matter WHAT documents are signed by who in order to presume to leave you off the hook for this, it is NOT going to stand up in court if there is no precedent for it and if the judge decides to rule in favor of giving the custodial parent at least a part of your and your wife's future income. As you are undoubtedly already aware, they ALWAYS rule in favor of the kid, whatever that comes to mean in any specific case.
Your call but you'd be better off going to Vegas every week for the rest of your life.
Having said all of that, I didn't get the impression that they wanted you to plow the field and plant the seed personally but if so and if your wife is okay with that, Wow. Just, wow! Send video!
As for jealousy, your wife could easily change her mind and despise you for it later on and so could the partner of the other woman. Strange things can happen in those cases.
Frankly, I think you have more cons than pros on this one.
--Wag--