A tree is pretty and hard to resist asking for if there is any option at all. We do one because my Dh is a (disillusioned) Christian but still enjoys his traditions. I don't care for it, but I'm not sure that relationships are all about no one being uncomfortable. Dh comes to synogogue with me and Dd, sits through long services, supports our keeping Shabbat/Sabbath with many restrictive/freeing rules (depending on how you look at them).
The previous owner of our house left some gaudy exterior decorations in the basement, and this year Dh wants to put them out this year. I think they are tacky if not just plain too Christamassy, but really, if it will make him happy, he's the guy who puts on a kippah when he needs to, and listens to me practice chanting torah day and night (It's an acciction of mine), when he'd rather be listening to the Rolling Stones. We recently set up a kosher kitchen and he puts up with that!
I grew up with a Christmas tree because my parents were children/grandchildren of immigrants and a tree was a sign of being American. I don't think they had any idea about all the Christian heaviness of the tree. Having a tree made me no less Jewish. Even though now, in my own home, I'd rather pass on it, we do it, and put a few gifts underneath because it makes Dh feel good. Most of our gifts come at Chanukah, not that that's not really American commercialism, right?
Biggest of all, here is a guy who is the son of a minister and a church lady, who is enthusiastic about raising our daughter Jewish!!!! We have had our discussions about whether it would have been better to mix things up more (a while other thread topic), but in the end there are no regrets.
Having given my speech about sacrifice for your spouse/partner/family harmony (Shalom Bayit), here's one idea that might help you. As Dd grew up, Christmas was always Dad's holiday or Grandma and Grandpa's holiday, and we do it with them. Would you feel OK rephrasing to the 4 year old something like, you know, you're right, a tree is pretty and Partner grew up with one. You have a good idea, lets do one for Partner. Then a troublesome behemoth in the living room can become a gesture of love.
The only thing I'd do differently in my years of putting up the tree, is extracting a promise from Dh that he will HELP ME TAKE IT DOWN! Maybe as part of this you can get that kind of deal in your house. Good luck and Chanukah Sameach!