I think you need to escalate the "withdrawal" and spend even less time with SIL, so she has less time to drive you crazy with this. Also, I think the broken record technique is the only one you can use.
However, I seem to recall the last time DH was considering helping SIL with something financial, he decided to give her the money after you recommended that he shouldn't do it. So I would make my position very clear with DH. "Honey, I do not think we should help SIL with this bed issue in any way. She needs to go to her doctor and go through the proper channels to get the bed through the aid programs. Otherwise, we're teaching her that any time she has a problem, all she has to do is come to us with her hand out and we'll pay for it. If you give her the money for the bed after I've asked you not to, it will hurt my feelings and make me feel as if you're not taking my opinion into consideration."
ETA: Also, I would recommend against any position that puts you "between" DH and SIL so you are the go between. This is not a healthy dynamic. It puts strain on you. It puts strain on your relationship with DH. And it teaches SIL that she can get what she wants by putting strain on you and your relationship with DH. If that means that DH has to deal with SIL directly so be it. He has left you in charge of maintaining this relationship for too long.