Why do you say, OP, that you father doesn't love you? Don't answer, of course, if this is too personal a question. It's just quite a strong thing for a daughter to say. Your SIL is being self-centred, not wanting to know the reason for your difficulty with your BIL; how convenient that she does not have to attempt to comprehend anything - that appears to be more for her own selfish reasons and wanting the both of best worlds to suit her, rather than the philosophy that it is none of her business and is between you and your BIL.
I don't know your background, obviously, and I never felt that my father didn't love me, but I, and others, had an often difficult relationship with him. At one time, I had little contact with him for several years, but then there came a time, within myself, (and, yes, it was Christmas one year) when I really didn't want to carry on like that with him, regardless of anyone else's attitude; it was what felt right for me as a person to do - so I initiated contact, and he came for Christmas, and I and my dc, had five more happy years to remember with him before he died. I am so glad I did that. It comforts me since his death.
I realise my situation is different to yours but it took quite a step on my part to make the contact, and the reasons are different obviously to yours. I understand you have tried and got no response yet, but you have made an attempt, and ultimately, we all have to make the decision to do what we think is right for us and our families.
Christmas always brings all these family issues to the forefront. And, we have the urge to make things better, improve things, lay bad feelings to rest. And, tackling them, as you were, the other day, feeling rotten, does not put you in a good place anyway. My family situation is still complicated, but some of it is beyond my control, and that is other members' responsibilities.