As a PP said, if your divorce isn't finalised you're moving a bit quickly here.
I don't understand why you can't split the household bills properly - even with your explanation.
This is about more than just money though.
For example pocket money - I would rather give my child this, than treats, because it's my parenting approach that it's part of the building blocks of teaching them about saving, budgeting, and value for money. Since my 7yo started to have it, I can say "yes you can have it - that's like a whole month of pocket money though". And even with the money in her hand (say it's a birthday) she'OK sometimes say "I love it - but no way is that worth a whole month!" and leave it. If her stepfather (to be, my fiancé) said he didn't want to do pocket money, I wouldn't budge on it.
It's a bit easier for us because he has late teens and I have a primary age. I spend more on his than he does on mine, because I earn more. In a few years time it'll just be from "us" but right now we don't want them to feel their dad just became a couple unit, full stop. He's widowed, so we're sensitive to that.
So the Agee of your kids matters a bit I think.
Another issue is you say your kids don't appreciate things as much - then don't worry. Either that's bad (too expectant and so getting less is good for them) or it's good - they're not materialistic - but again, it's fine to give them less.
If you genuinely want to be a family, with small kids, you need to treat them the same. But that doesn't always mean equal money.
Have you talked this through for the long time?
My fiancé will do "well" from our relationship financially - he's letting out his house and keeping the profit and paying no bills at mine, because of our incomes. But we have both agreed that the money we have brought in as savings and will save in future separately, is earmarked for our own children only. You need to be aligned on this now, not later. If you think toys at Xmas is tricky to negotiate, wait til it's share of costs at uni. It's easy to give an extra £100 at Xmas to your SC. What about when your half of uni halls is £5K more than you would have spent?