TEll me to sod off if this isn't helpful and I know it's probably the last thing you'll feel like doing when sleep deprived but I find that having to get up to get ds1 off to school has helped me deal better with ds2 and particularly dd.
When I just had ds1 if he had a bad night but would sleep in the morning like you described ds2 doing this morning, I would sleep.
When I just had the ds's I would get ds1 fed, dressed etc and send him off to school (opposite our house so I didn't need to leave too) then have a lazy morning in our pj's
Now I have to get ds1 up and out and ds2 is up as well as he has to give ds1 his school bad, lunch box and school coat every single day(he's 2.5, I think my youngest are around the same age as yours?)
So even if I've had a terrible night with dd I stomp around upstairs until she's awake and carry on (as much as possible) with our usual daily routine even though I feel like staying unwashed and undressed lying on the floor sobbing "Why me??" I get us all dressed and make sure we're out of the house by 9.30. I take ds2 for a walk to the shops and a run through the park and dd usually goes to sleep in the pram.
We're back home by 10.30 and then I manhandle the pram a bit to wake her up, scrunch the raincover type things and then give her a feed straight away.
THen the ususal playing, washing, snacking (I find she loves sitting on the floor next to ds2 for a snack, she copies everything so try and use it to my advantage)
Lunchtime and feed
And then back in the pram for another walk about 1.30/2 o'clockish.
Home by 3 in time for ds2 and repeat the waking thing again with another feed.
Between 4.30 and 5.30 I'll keep offering her a feed but stop if she looks like she'll sleep, put her on the floor, pick her up and try again.
Teatime, bathtime, supper (i'm a big believer in supper) play and feed by which time she's usually knackered. Then I'll spend an hour feeding her and then she ususally goes off.
I'm not saying this would work for you but the controlling off her naps has definately helped me (at least I feel like I'm doing something about it)
And I make sure I have lots of snacks and drinks throughout the day to stop me flagging too much.
I totally empathasise with you, ds1 didn't sleep til he was 4 and he still doesn't need much sleep now, so I know how much it affects everything
If you are able to take up VVV offer I would! Or at least do the whole evening settling yourself and then let DH take over during the night.
When DP was going in to settle DS2 down I would put my head under the pillow and cry, I find crying babies cause my milk to let down and I'd feel like I was denying him something that he needed but then he would just stop and let DH rock him for a while and finally settle down. It wasn't the easiest week in the world but when he started to sleep for 12 hours it was all worth it and the change to our family's life was worth it to, living again not just head down and get through another day iyswim?
Anyway, chunnered on for long enough, I really hope you find a way of dealing with this that you are comfortable with and that works for your ds2