I think you've got to fight the battles you're likely to win.
For example, DH steps up for a week then it goes back to 'normal' again. Can you say to him you need this Saturday or Sunday to yourself and he needs to take the kids out for the day or do something with them at home. It doesn't matter if they spend the day on the PS4 and have pizza and sweets. You're getting time off and a bit of a 'junk' day didn't hurt anyone now and then. Tell him when you need him on duty and let him do it his way. As long as they are safe and fed that's all that matters. If you have to tell him every month, 6 weeks or whatever so be it. As long as he does it, a reminder from you doesn't matter. But don't use your time to do housework or whatever. It's your time - go to a cafe with a newspaper, go for a quiet walk, go to your galleries, go window shopping - just enjoy some time.
Can DC1 play with DC2 for a while? I know you said there's resentment so you may have to supervise but just sitting back and letting them figure it out with a bit of intervention when needed can sometimes be good for everyone. They may even find they enjoy each others company and that's a bit of time you're clawing back eventually. Could DC1 teach DC2 how to play chess? Even if you're reading a book quietly in the background you're not the centre of DC2 world at that point so they're learning it doesn't always have to be you.
Eventually that becomes a thing the two of them do instead of you.
As for the kids moaning when you take them places. That's normal. DS did that. But we used to work hard at making it fun. Going to a gallery and choosing our favourite piece. Working out what the man in the painting may be saying to the woman in the painting (the funnier the better as it got DS giggling so he saw art as fun and started looking forward to going to these places). Going to a museum and getting him to think about how a thing was made or how the people lived or how that dinosaur would eat or move. Anything to make things fun. Then teaching them to quietly think/read/look - how to take things in. We'd say something like 'there's a bird in that painting - can you find it' or 'what did that sign say about Churchill's wife? Go back and have a look' It's hard work at first but eventually DS started going off to areas alone (in sight of course) and telling us things he'd learned about the things he was interested in.
DS now has a huge interest in history, geography, astronomy and science and suggests places like Greenwich, Natural History Museum, Chartwell and Hever Castle etc for places to visit as he enjoys them. When he was 9 years old we went to Hadrian's Wall and the Roman sites along it - the stuff he learned and taught us was amazing and the way his mind started to figure out how a shoe may have tied or a piece of equipment might have been used was incredible!
Kids don't automatically enjoy galleries or museums - they have to be taught how to immerse themselves and get enjoyment from the place.
So if you go tell the kids you're going for an hour, choose a couple of things to look at and get them involved. Have a look on line and choose an area together if it's a museum. Just do that area unless they show they're wanting to stay and do another section. DS was (and still is) fascinated by the huge gates and statues in the British Museum and he'd make up stories about why the gates were needed, who the enemy was - normally aliens from another planet with blasters but who cares? It's fun! Then go for a McDonald's or get a cake as a treat. Make galleries and museums part of a fun day rather than a traipse around. Short and sharp, straight in and straight out then gradually increase the time.
You have two sources of help in your home - your DH and your DC1. They both need to learn to step up to help with DC2 and DC2 has to learn to value them as much as she does you.
Sorry for the novel!