Firstly, you DO resent this baby because of your SIL whether you want to accept it or not, it's why you don't and can't love your DN. That is what's shocking.
'I am nice to my niece. She wouldn't know anything was up'.
Kids always know. I have an aunt that's nice to me but I definitely know she doesn't like me or approve of my father.
'They struggle with money because of this and DB is working so hard on over time plus extra night jobs. He is exhausted and still comes home to do a lot of the bathing, feeding, cleaning and bed time. I don't want to delve in to this too much but I stand my ground on this one, no, her day off isn't well deserved.'
He sounds like a good father, getting quality time with his children whilst he can, whether that is forced upon him or he chooses so. It's definitely not your problem or business what they do. If he works overtime and night jobs she's caring for a child, you are hearing one side and blindly believing that to be the full picture. If someone asked my DH similar he'd say he does lots but he does average with a full time and part time job whilst I have been at home 24/7 with a cosleeping baby and no break at all. Whether I have done chores myself does not come into it as I've been raising a child on my own whilst he works. How they divide their labour is up to them, as family you will hear the worst but rarely the great things because we all need to vent sometimes.
had niece not been there, we would have had a very different (easy) day'
You just don't want to be around her because it's too difficult for you but if you tried more, perhaps in months to come you wouldn't feel the way you do.
'My DCs were looking forward to our day out but are lovely enough to understand.'
No, that is basic decency which you should teach them anyway. They should spend time getting to know their cousin because they, too, will pick up on this
'What some of you are saying, I don't deserve my day off. I was with an upset, crying baby that I had to soothe, which was hard work'
You deserve a day off but your DN also deserved that day with her DGM and is entitled to it as are your children. You wouldn't find it an issue if it were your other DN at that age, because you love her but also because she would have been more at ease where you put in the effort to spend time with herm
Lastly, your DB has told you she's having a "day off" but perhaps she was going to a serious appointment she didn't want to tell you about. I wouldn't tell you either!
I don't get on with my SIL but always offer to look after DN because I love her which has come from making an effort and not being able to deny love to a small, innocent baby who is not responsible for her mother's behaviour. Spend more time with your DN, work on being a better person by improving the relationships around you. Perhaps you could try to extend it to SIL too.