I grew up in an area where Christmas was everywhere but as a Muslim I didn't celebrate it. It never affected me negatively. My parents clearly told me that Christians celebrate xmas (well it was like that 20 years ago but here in the UK it seems like less of a religious thing and more if a commercial or even just December holidays type of thing). Your children will understand why you don't celebrate xmas if you let them know why.
My Christian neighbour told me that Christmas isn't even from religion and has a pagan background. You'll find loads of videos on YouTube about this which will help strengthen your own belief as to why Muslims don't celebrate xmas. You might think it's no longer celebrated that way but it is.
You can still enjoy the December holidays as it is. Kids get two weeks off of school, it means more time together as a family and meeting up with friends. In all that as a kid I never felt I was missing out on decorating the house etc although I understand that is the culture you come from. You can still enjoy the holidays without linking it to Christmas.
There is a talk on YouTube by Yusuf Estes on Santa Claus. It might give some insight. He was a priest before.
I think Eid is becoming very commercialised with the expectation of gifts too. Islam really is about gifting randomly and not limiting it to birthdays, Eid, Mother's Day etc. I mean how many threads do we see on here and other forums where people are forgotten about on their bday, mother's Day etc? Islam is about embracing each person throughout the year. People are forgetting this though. Gift randomly. It means so much more.
With Christmas lunch, we just don't name it anything. We eat together anyway but around Xmas there are more people around as more people are off school etc so it just means more people and more food but once again there isn't really a need to label it anything.
Basically you can eat all you want but don't need to label it with the Xmas name. It can just be eating because there's food and more time or gifting because you haven't seen x person in a long time but gifting your kids an Xmas present wouldn't be a part of Islam.
It's like how people from Indo Pak backgrounds have traditions which are 100% not from Islam and actually from their culture which might be mixed with a major religion in the Indo Pak world. It can interfere with ones core belief in Islam which is no associating partners with Allah (therefore disassociating oneself from the festivals) or another example is forced marriages or women being treated worse than men and expected to be slaves to the in laws. This isn't from Islam but from some cultures from back home for certain people. Islam is very much about creating clear lines and a clear path.
We used to see Santa at the local shopping centre and his elves would be giving out free chocolate coins. We very well knew he was not real and was just a man dressed up in red clothes, we took part in the activities at school etc. Your children will not be affected by not celebrating Christmas.
Definitely check the Yusuf Estes video.