I think the issue that's being missed here is that this isn't an equal situation or a direct comparison with whether it's "ok" to take a daughter out for a "treat" but not "ok" to take a son out or whether it's "ok" to take any of your grown adult children out without their spouse. If I had to hazard a guess, I'd say you (and anyone else saying this should be viewed the same as a mother taking her daughter out as a treat) either don't have daughters, or were raised to believe that a woman's place is in the home and a man's place is to be the provider and occasionally "help out" with the children and home tasks if he's not too tired after his uber-important and terribly stressful job. The ingrained misogyny in society is perpetuated by women having these kinds of attitudes and modelling them for their own sons. In these kinds of situations, you often find that the man's work ends when he gets home and the woman's work never ends because she's never off the clock, even when her partner is home.
You often hear people (almost exclusively women) say "oh, he's so good with his children" and "oh, he's wonderful, he does so much to help around the house". My own mother says this kind of nonsense. I've literally never heard anyone saying the same sorts of things about a woman, and frankly, it would sound ridiculous for someone to say that about a woman. It's time society understood that it also sounds ridiculous to say these kinds of things about men. This is a period of such a huge adjustment for your DIL and you went and pissed all over it. She literally made a human and birthed that human, and is now dedicating her every moment to looking after them, the home and probably to a large extent, supporting and taking care of her husband's needs too. Meanwhile, although he now has some additional responsibilities and the stress of providing for the family financially by himself as your DIL is currently a SAHM, he hasn't given up his entire identity and every waking moment for his family and if anything, he's gone up a notch in society's eyes, whilst his wife has become less visible as an individual,
He even thinks it's perfectly OK to take himself away during some of that precious and small amount of time he has available to spend with his new family. Not very surprising, since you obviously brought him up to think that way. SAHMs rarely get time away out of the house without the baby. Her husband, your son, is at work for probably at least 80% of the waking hours in the week, perhaps even more, whilst your DIL is with the baby, oftentimes on her own, and based on your offer, his acceptance of your offer and her response, I'd be willing to bet she still does most of the work even when he is home because otherwise he wouldn't have felt like he was free to accept your offer. I remember watching the clock and being so happy and relieved when I heard my husband's car in the drive every evening so I could have just a moment to have a shower and get myself something nutritious to eat, or have a nap. I got more and more excited as the weekend approached so I could get a bit of respite and spend some time with him before I was all alone again on Monday morning. Even though I still did at least 50% of the parenting at the weekends, just having the support and the ability to hand the baby off was very much needed.
Not to mention that my husband wanted to spend as much time with our daughter as he possibly could and would never have dreamed of going out with his mother or anyone else during those early months as a new father (nor would she have dreamed of asking him to). Admittedly, our daughter wasn't an easy baby and we had her during lockdown so I couldn't go anywhere with or without her, but still, 6 months is very early days and it sounds like your DIL might be feeling something similar and I do find it strange he even wanted to take you up on your offer. When he is at home, I wonder if he spends any time with the baby completely by himself whilst DIL has completely uninterrupted time away from caring for the baby or your son's needs or doing other tasks around the home? It sounds unlikely. Although stressed at his work, and doing lots of overtime, your son is at least getting time to be himself, with his own identity, away from the other, just as stressful demands of parenting., plus probably getting some down time of his own. Meanwhile, your DIL has just gone through the transition to motherhood, and in spending so much time by herself at home with a baby, has possibly lost part of her own identity and is now viewed by society, and possibly even by you and her husband as a unit with the baby rather than as an individual person with her own wants and needs.
At this point, his respite from the duties at home IS his work, at all other times, he should be sharing (and should want to be sharing) the load with his wife. Her respite is what? In the opposite scenario, it would be an entirely different thing for you, or her own mother, to invite your DIL out for some time away to provide some relief for her than it is to do the same for the man who is already spending a great deal of time away from his home, his baby and his wife. If you felt the need to reward anyone for their hard work (which as people have said, is a bit weird anyway), you could have offered to take them both out for a meal or you could have offered to babysit so they could both go out on their own as a couple. So whilst I think that yes, you were out of order for inviting the working father of a 6 month old out by himself for an extended period of Mommy-son time as a "reward for all his hard work",
it was also out of order for him to think it was appropriate and agree to it. And to those people who say he should have checked first with his wife, even that shouldn't have been necessary, he should have already known that in their dynamic, this wasn't the right thing to do. He should have known his wife would be upset by both the offer and the way it was described and declined your offer before even presenting it to his wife as a viable proposition.
I also find it rather strange that he would even want to spend such a large amount of his free time away from his wife and baby at this time, particularly as he's working so much overtime at the moment, but clearly, you brought him up this way. I think it's great you have the sort of relationship with your DIL that she felt comfortable to communicate with you in the way she did, and I also think it's great that you are asking for and open to outside advice. I think she was a bit out of order to demand you babysit to make up for it, but the actual conversation might not have gone exactly how it sounds.
You also said she said "she's feeling like I'm more invested in my son's life and that I lean more heavily towards him, which is making her feel unappreciated and overlooked" so this is obviously not the first time you've effectively dismissed her or viewed her as secondary or less important than your son and that there's been a pattern of you "bigging up" your son and his role to the detriment of your DIL. I get that he's your son and you're bound to have a bias in his favour, but he's a father now, and you're not helping the stresses in his or your DIL's life by taking him away from his family when his time with them is already so limited. I think you might might have a little work to do on your own internalised misogyny and once you truly understand why it wasn't appropriate for you to ask (in the way that you did) and why it wasn't appropriate for him to accept, perhaps you'll be able to help him to understand this as well. Oh, and you'll also be able to apologise, unreservedly to your DIL and hopefully repair your relationship.