Anyone who would use the phrase "gay lifestyle" isn't worthy of your time, even if that person is your father. You need to do what's right for you - and that may come at the expense of your relationship with your family. I'm glad your mother is accepting - somewhat. Mine is grappling with it with a somewhat lukewarm response. I haven't yet told my father, and I don't know how he, or my stepmother will react... I've been dropping hints too. It's only a matter of time before comments are made about my shape changing, and so on... you've got to do what's best.
It's a good thing you're seeing a specialist and getting the support you need. You need to prepare yourself for it backfiring in your face, however. Your father, who is already uncomfortable with you, might just be pushed over the edge... prepare yourself. Ultimately, your happiness matters most.