I don't think you actually did anything wrong. I think your DS did.
You said he's been under a lot of pressure at work and he accepted this invitation to go out with you on a Saturday. Your DIL complained to you that your DS going out with you made her life harder.
Reading between the lines, your DS is currently working longer hours than normal (meaning your DIL has more solo parenting than normal) and your DIL probably doesn't know when he's coming home each day (because your DS doesn't know either). He's probably more tired than usual, so the time they do get together isn't just shorter but poorer quality.
I suspect she feels unsupported by your DS.
Has she voiced this to him? Probably not, given he accepted your kind offer to go out on a Saturday. If he knew - and he was a good partner - he would have told you he'd have loved to go out with you but he knew she was having a hard time parenting at the moment so he'd rather do something with her and their child.
There are many reasons why she wouldn't have voiced to him, and why it has instead come out in a grumble to you. Maybe she's too reliant on his income. Maybe she feels insecure about being SAHP. Maybe she does think he's working hard and doesn't want to make him feel devalued, and is instead taking that hit herself.
But the fact you now know how she is feeling is really positive, because you can give your DS a heads up before that nagging resentment on DIL's part turns into a massive wedge that potentially breaks them apart.
You're not wrong to want to treat your DS and spend him with him one to one. But given the age of your grandchild and the pressure that both DIL and your DS are feeling... for now, the kindest and best thing to do is to try to facilitate time for them as a couple, so they can feel strong as a couple, and that helps solidify that family unit for your grandchild.
The baby won't be tiny forever. There will come a point where you can make a kind offer to go out with any combination of DS, DSIL and/or the child without anyone getting frustrated!
You were kind, and I don't think this was about you. Many DILs wouldn't have trusted their MILs enough to give them that insight into their actual feelings. Rather than feel like DIL has tainted your lovely gesture for your DS, I'd see this as a really positive thing to have happened. Not only did your take your DS out for a lovely day, but you can now plan to offer babysitting time more often (when you can) and that you know that's a gift that both your DS and DIL really need and will cherish. If DIL hadn't been in a position to tell you how she was feeling... maybe no one would have ever known until it was too late.