Tyrion has served as Prime Minister and Minister of Finance for the Lannister regime. He knows Joffrey is not the rightful king, yet continues to serve him because Joffrey is the Lannister king, going to so far as to burn hundreds if not thousands of men alive with wildfire in service of that King. Tyrion married Sansa because he as told to marry Sansa. As far back as the second episode of Season 1, Jaime all but confessed to pushing Bran out a window and Tyrion responded that of course he wouldn't do anything about it. Tyrion occassionally poo poos his famliy's behavior, but he never does anything about it. At best, he's an enabler and after-the-fact accomplice.Tyrion's offer to help Sansa was beyond obnoxious. He can't bring her family back from the dead, and he can't and won't bring their killers to justice, and both Sansa and Tyrion know it.
Personally, I don't think that Tyrion's inactivity means that he is siding with his family. I mean, what can Tyrion do knowing that Joffrey isn't the rightful king? Or, if they had lost the war, what would have happened to Tyrion? This is not saying that I'm necessarily concerned for him, but how many people in series care that Tyrion hasn't done anything to harm anyone or doesn't agree with his family--very very few. All that matters to them is that he is a Lannister. So, yes, he wants his family to win the war because the little protection he had is gone if they lose. Can Tyrion defy his father? He can crack smartass remarks, but all of his security is because of his father. Without that, he life looks pretty grim. Regardless of what Tyrion feel, his family is going to do what they want to do.
Regarding Sansa, with all that was said, Tyrion does what he can. He's alive and stays alive because he is smart and knows when to meddle and when not to meddle. I really don't see what he is doing or is trying to do for Sansa obnoxious. Even if he can't do anything for her, does that mean he should do nothing? Leave Sansa to her own devices and call it a day? What people are saying is that some sort of alliance can be formed, so that down the line, Sansa can get justice. It's not going to happen right away, but it could if Sansa learns to keep her anger in check. Is she any better where she is now? Quietly seething and depressed? At least Tyrion has resources even if they are limited.