A few thoughts thoughts:
First, my husband and I do MUCH more beancounting than you are describing here, and we are extremely happily married. I think you're actually doing great in balancing the realities of the inequality that's playing out right now with a recognition that the goal is to get to a balance that works for you, not some exactly equal division of time.
Second, since you are doing more of the childcare, maybe he just needs to use his non-working days to do more of the housework so that you can have a few hours to putter around after the kids are in bed. I tend to think that it's good for our family dynamics to have equality in childcare, separately from the issue of overall household labor--i.e., we try not to get into a pattern where one person does all of the childcare and the other person makes up for it by doing all of the housework, because we both want to have time to nurture our relationship with our kids. But sometimes it can't be helped, and if you're burnt out, you don't want the perfect to be the enemy of the good. It sounds like he's already doing bedtime at least two days/week, which is good - that might also be a time when, instead of being out of the house, you might try to come home a bit earlier and spend the time doing your projects.
Third, is there any chance that the family who is willing to babysit would be willing to host a sleepover at their house, at least occasionally? My parents actually love having my kids sleep over at their house, and it's a great adventure for my kids. We generally use the time to have a luxurious morning of sleeping in, but you could get up and do some DIY or cooking then. If you're family's not genuinely enthusiastic about it, then it's obviously not something to push--but I know that in my case, it's a win-win.
Alternatively, maybe you could alternate getting family to watch kids for a date night, and having them keep the kids for an afternoon where each of you do solo projects. Both are important. And if you start using a gym with childcare, it sounds like you might be able to get some weekend time together while you exercise. My husband and I will take our kids to the Y and exercise separately, but then sit in the sauna together for a half hour or so before we pick the kids back up. It's not a ton of time, but it's time when we are out of the house and relaxing, which is hugely valuable to maintaining our connection.
Finally, I do think it's worth being realistic about the amount of "free time" that's available in life with young kids. Y'all are actually both getting more kid-free time than my husband and I do. Some of that is by our choice; we prioritize family time on the weekends more than y'all do. That's not a better choice, just a different one, that works for our family and our needs. If y'all need more solo time, then you should absolutely prioritize that. But I think we'd have trouble building in as much solo time as y'all have right now (21+ hours minimum between the two of you) while still doing the basics required to keep our life together (cook, clean, pay bills, do repairs, etc). If we did, it would come at the expense of basically all our family time and couple time. Again, if you can make it work because you have family support, using gym childcare, etc, that's great. But part of this equation might have to involve your husband admitting that the amount of time he expects to get is just not realistic.