Would you let him file with your child?

So my fiancé and I along with my three kids are living together. The house belongs to him and his ex (kids mother).

He’s been struggling with work due to the weather and has gotten notices of foreclosure and his truck being repossessed, cable being cut etc. We haven’t been on the greatest of terms due to dumb things like him texting other women, disappearing for days, etc but lately he’s been trying to be a little more nice.

I’m going to file taxes soon and he is as well but he doesn’t really get anything back. Definitely not as much as I would. So he’s been wondering if I will allow him to use one of my kids socials to possibly help his tax situation. In the back of my mind I’m like hell no but we do live with him. Also if he was to lose the house or anything I could always go back to my moms. Idk if he would use the money to help his ex (I think he pays her car insurance) or basically use me for this help and then go back to treating me like crap. What would y’all do in this situation. He keeps saying he’s talked to his “tax guy” and all I have to do is give him the social security number and that’s it. He’s sketchy so idk.