My husband and I are in a blended family as well. We have our kids every other weekend, but get mine back at 10:45 because my ex will drop them off at our church specifically so they can go, which I am VERY thankful for. Our parenting plan says he can have them until 6pm. My husband ex gets their daughter the entire weekend, we don't see her for 2 weeks straight.
So my point is I can understand where you are coming from, although my husband does attend with me. There are several women who attend our church with their kids faithfully whose husbands are not saved or choose to not attend. I think you should bring the kids and attend anyway. It's a GREAT way for the kids to make friends outside of school and the neighborhood and the same for you as well.
I hope this doesn't get too long, but I will tell you a couple stories about tithing personally. My husband and I have been together going on 6 years. We both are 100% commission only, check to check and each month we have NO idea how much our income will be since it depends on our sales. Ok, so we are sitting in a service at our old church. Pastor is speaking of tithing. I lean over to my husband and tell him I feel like I should tithe everything in my wallet. He says, how much? I say, around $88. He's all, how about $20? (lol, no kidding!) I said, no, I really feel like I should give all of it. He says, ok. So I do, even dump out my coin purse. So I'm feeling great and we go out to my car, hop in and my gas light comes on. I immediately start bawling, hubby says, it's ok, we'll make it. Ugh. We literally had $20 in our account, no kidding. So he leaves Monday morning for work, decided to stop by one of his clients offices, they say, hey, we have 3 new employees you need to see...he sees them, sells them and we get $1200 deposited into our checking account the next day. In 15 years of sales for the same company, he has RARELY had this happen.
Now at our new church, the Pastor gave us a challenge for 90 days to tithe 10% of our FIRST income. Meaning, before you pay taxes or anything else. That Sunday night, my husband and I decided to do it. That next Monday morning, I woke up, logged onto email and was basically fired. 2 days later, I get an email from my ex saying he has been on disability for 2+ weeks and won't be returning to work for a while so I won't get any child support. The next day my husband gets an email from a client cancelling a HUGE opportunity that likely would have been 5k at least in commission for him. A month later, I have a weird health issue that costs us dearly in copays and premiums. I mean over 10k already and they don't even know WHAT happened. All of this after we committed to doing what is right. But guess what? Since Feb, we have been living SOLEY off my husbands income and only the last month did my ex go back to work so I've gotten 2 c/s payments since before Feb. So living on ONE income and STILL tithing 10% off our first income, we have lived the same, if not better than before. In fact, we just found out my husband qualified for the NATIONAL company trip to NYC this year. He has not qualified for that trip in the 5+ years we have been together. Interesting how that works...right? God is not going to let you go homeless or starve.
Sorry this is so long and if you haven't given up on me yet, here is my 2 cents. You said your husband "believes in God", which is great. So if I were you, I would be honest with him in that if YOU work and have an income, tell him you feel you need to tithe 10% off the top because it says to do that in the Bible. (I can't remember the exact verses but message me if you want them and I'll find them or ask my friends.) Do not stress him about going but you go with the kids. Go to everything you can and get the kids involved in their groups and you get involved with the women's group. Keep praying for him and invite him to all the "parties" they have. Like this Saturday is the mens/womens breakfast. If he says no, go anyway. But keep inviting him. I'm pretty sure one day he will see the positive changes in you, and when you say, hey, there is a church pool party that would be fun if we all went to, he just may say yes and go.
In the event that you are NOT ready to tithe and/or feel it would put too much pressure on you guys, then just go to church. Someday you may feel differently and start tithing but in the mean time, make sure you at least start going. We all LOVE to go and are very active. I wish the same for you and someday it may happen. Good luck.