The alpha theory is flawed and in fact was disproved by the person who came up with it.
The partnership between you and your dog can be much more equal.
He wants sex, you do not, so you refuse and vice versa.
Most animals will come up with some way of asking for sex. And you can of course ignore the request and refuse to have sex. The same way if you come up to your dog and he refuse to have sex with you in that moment.
You are not going to be your dog's bitch because this is impractical and in fact incompatible with normal life.
You are going to turn your dog down in public, because you are not a moron, and you are not going to do dumb shit like have sex with an animal in public.
That is if at all your dog will ever ask for sex in the public. Which my dog has never done ever in the past 10 years despite being hornier than I am.
At best he asks me for sex when we have visitors over or when I am visiting my family. And all you have to do is to take him off you and ignore the request.