Emm I am so sorry for you.
Maybe the time is coming where somebody (you? DH?) has to sit down with your son and have a heart to heart. This atmosphere s not doing anybody any good and frankly your DIL comes across as a grasping, scheming individual. Do they both have jobs? (I may have missed that) Is there any specific reason why they are particularly hard up? (I did read about their extravagances) If it is just because they are feckless you will have to harden your hearts somehow - not easy. If they are keeping to a strict budget, of course things are different, but I suspect not.
Going out for breakfast sounds like one to bin right away - get there later in the day. Do lunch and choose your venue(2 for 1 offfers are so often available) If questioned just say chirping Oh money doesn’t grow on trees you know.
Whatever the coolness which may ensue, you must stick to your guns, at least you have each other for moral support.
Constant pleading poverty is very depressing and you may have to “act” jolly and breezy and deflect these unsubtle attempts to get you to part with more money.
Finally, regarding baby gear, IKEA do an excellent cheap high chair - everybody I know has one for their grandchildren and you could at a pinch get stuff from local mums’networks selling toys etc. I am NOT saying buy new for your DIL, but choose your battles. It may be that you offer a compromise, as in, “we can only buy you ONE out of .., which would you prefer? “ Do not fall for the “travel system”which usually costs as much as a small family car, by the way.
Good luck.