You're overthinking this. Approaching Beesbury wouldn't be an option if his support wasn't likely to be efficacious. Lannister, Westerling, and Mellos have all been excluded as options, and they're all equally on the Small Council.
Maybe but we really need to fully convince him for him to throw his weight behind us.
I agree with Zimmerwald, not only do I think this is very likely to succeed, but also, actually showing Otto in tangible ways that we can be a very powerful ally for him and his family will certainly help in our efforts when we ask him for an alliance to keep Alicent by our side.
"I will say you can absolutely use any bribes made now as part of an argument to convince Otto later."
I don´t like us going to him for this because i think we would be approaching him in a position of weakness and he would absolutely fleece us. We are not that good on intrigue.
It is other thing if we get RTA approved and then offer him a slice of the pie as an incentive for accepting us and Alicent.