I'm so sorry that Oreo isn't doing too well at the moment.
Of course we don't want her to suffer, but hopefully there will be another solution other than putting her to sleep. Is your vet rabbit savvy? It is possible that a vet who took special training specifically for rabbits would know more solutions to treat this kind of problem in rabbits. I googled prolapsed rectum in rabbits and found a message posted by a bunny care taker who has a bunny with the same problem yours is having now. I've copied and pasted what she posted, how she was handling the problem. I hope it can be useful to you.
"I had a litter of babies come in a month ago in which one had a prolapsed rectum. With pain relief and bathing in salt water I was able to get it back in and she was alright for a few weeks although in the meantime it happened to another of the litter. The vet put hers back in and suggested a dose of steroids if it happened again. The second one has not had a repeat, but the original one has. She has been on a high dose of steroids for a week now, every time I reduce them out it comes. She has had an op to place purse string stitches in which helped for a few days but once the steroid was reduced it came out again anyway and now the stitches have come out, yesterday of course. I got it back in yet again and have had to increase the steroids, but that is not ideal. She and the rest of the litter (6 in all) are on a course of Septrin in case it is coccidioisis and panacur in case of pinworm. The whole litter were very stunted, the rest now are growing noticably every day, leaving the little doe with the prolapse behind, although she is eating well and has normal lovely poo. She weighs less than 500g, the others just over. They gobble A&P and readigrass and hay and drink huge quantities.
The only ref I could find to prolapsed rectums in my rabbit books was in reference to coccidiosis, hence the Septrin. The vet said pinworms could cause it, hence the panacur.
It looks to me as if the whole litter have a slightly misshaped anus that lets this happen more easily although I think each prolapse is caused by swelling as the steroid has caused it to go back in on its own a couple of times. I suspect that infection has set in in the origional little doe (Tara) which is why it keeps repeating. It had been out for a few days before she was brought to me and was hugely swollen at that time. She initially had a short course of baytril, cautiously and not a long one as it is not good for imature animals.
She is lively and only looks in pain while it is out. As soon as I get it back in she is lively and happy again and eats well. While she remains so I don't want to give up. Has anyone else found a reference to this happening?"