Like, tell him you have been in a relationship for a while and things are going well and you have decided to move in with him because you're happy. Just make it sound as natural as possible. After all, he's your father and whatever he thinks of gay people, he would have to think twice before falling out with you.
But I think you should tell him tête-à-tête because if your whole family is there and everyone starts to openly defend you, your dad might feel under attack and react negatively as a defense mechanism.
Definitely the best piece of advice is to tell him. Whether or not he takes it well, you cannot control. You can only control that he hears it from you, and gets the straight goods, nothing through the grapevine, nothing off Facebook, etc. Then he gets a garbled story, and then it's going to be worse. It sounds like your dad is the kind of guy who respects people "being a man", so that's part of it.
If he takes it poorly, then he does. Your family will likely work on bringing him around, but it's better to know. I mean, you gotta be you. Regardless of the outcome, I think everyone here just hopes its for the best for you.
I think these two snippets of advice are the best course of action, at least what I personally agree with.
You may or may not be aware that I am a lesbian. My mother is very homophobic (and Catholic and Mexican and generally conservative) and I kept quiet about it for years to her. And it ate at me on the inside. Especially when I started having relationships with women, and I couldn't even say or hint at it or anything. Most of all, I (then as now) just want it to not be a big deal.
I ended up telling her about 4 years ago. Unfortunately, it didn't go well. Still today, it's a topic we rarely discuss. However, I'm glad that I told her even though it more or less was a total disaster (you know the worst reaction you hope your dad doesn't have but there's still a chance he might have?), because I was true to myself and honestly, the burden is on her to deal with it, instead of on me whether to tell her or not. So, even if the outcome is the worst possible in terms of reaction, I think you will feel better getting it off your chest. Plus, it sounds like you have a supportive network in your mom and sisters. Luckily for me, my dad and grandma are totally cool with it and this helps as well.
Good luck and a big gay hug!