If you and your parner and your kids go to WDW, how would anyone know if you were gay or hetero? Unless y'all were swappin' spit or hanging all over each other, you would like like any other dads and kids. I don't think you would have any problems at WDW if you behaved as you would expect other to behave, like you said. I have seen many men with kids in the parks. I don't know if they were married and the wife was at the spa or not with them, I don't know if they were divorced and it was their turn to have the kids, I don't know anything just by seeing them in the parks. I have seen lots of men in the parks with out kids too, and it never occured to me that they might be a gay couple, just as it never occured to me that they might be 2 guys that like Disney. Sadly, I know that not everyone see things the way I do, and you are correct about the more vocal right anmd christian support on the DIS, but remember there are all kinds of people in the world. If you take your kids to WDW with or with out another man who may or may not be your partner and may or may not have any kids, there will be at least someone who thinks that there is something wrong, even tho they have no personal knowledge of your situation.
WDW is probably one of the most diverse areas in the tourism world, I am willing to bet that if you and your partner and the kids go, it won't cause any odd looks. The majority of people at WDW are there to have a good time, and don't really pay attention to what is around them anyway. Hey, maybe I will see you there one day, the light bulb will go on, and I will think "Hey, that might be the guy who was worried about beong in WDW with his family." But knowing how I am usually not paying attention to the crowds around me, I don't think it will happen.
PS, there are a lot of Gay and Lesbian Cast Memners in WDW, too.