Did the vet take any head or chest xrays? With the rabbit favoring soft foods and having a suspected case of an ongoing upper respiratory infection, it seems likely the infection is either dental related and causing mouth pain while chewing harder foods, or spread to somewhere in the head that causes significant enough mouth/jaw pain to affect the rabbits appetite. Destruction of the nasal cavity from the ongoing infection, or infection spread to the bone, are a few possibilities.
https://www.medirabbit.com/EN/Respiratory/Bacterial/URI.htm
https://wabbitwiki.com/wiki/Snuffles
https://www.vin.com/apputil/content/defaultadv1.aspx?pId=11131&id=3843938
If the antibiotic azithromycin hasn't been tried yet, that might be something to consider. Though in rare instances some rabbits can have severe negative digestive reactions to it, and it can't be used in those cases, for the most part it doesn't cause significant issues and can be a very good and very effective antibiotic for rabbits because it penetrates the bone and tissue so well to get right to the infection, and usually relatively quickly.
I've had really good results with it treating various stubborn infections in my rabbits and really only saw some mild stomach cramping for a couple hours after administering the dose, then my rabbits were fine and resumed eating normally. Usually after only 2 or 3 days, I saw significant improvement with the infection clearing up visibly (and audibly), almost completely.
But it does depend on the bacteria present, on whether it would be effective. And for treatment, only the higher dose (50mg/kg) for an extended treatment period (3-4 weeks) worked at getting rid of the infection completely. The lower dose of 30mg/kg for 2-3 weeks just ended up in the infection returning after the antibiotic was stopped. Given on a semi empty stomach seems to work best at preventing a negative digestive reaction from occurring.
https://vcahospitals.com/know-your-pet/azithromycin
Since your bun is favoring soft foods, you could try soaking the pellets in warm water to make a pellet mush, and see if your bun will eat that on his own from a dish. Or use critical care or other recovery food mix, just more expensive. It does need to be made up fresh and shouldn't be left out all day (bacterial growth issues). If your bun won't eat the softened mush on his own, you may need to start syringe feeds, which is something your vet should have already advised on if they're a decent rabbit vet.
https://wabbitwiki.com/wiki/Medicating_your_rabbit
https://rabbitwelfare.co.uk/syringe-feeding-guide/
Garden/store bought greens alone (excluding foraged plants) aren't enough nutrition and your buns health will start to be affected. They're usually too high in water content, and too low in indigestible fiber and protein, to provide adequate nutrition to a rabbit long term. Fresh grass and forage are possibilities to supplement with if you have access, though these do need to be introduced gradually and one item at a time. I would also suggest to offer a salt lick until your bun is eating pellets or a pellet mush which has added dietary sodium, as only greens with all that excess water content, may cause a depletion of sodium levels in the body due to the excess urination.
https://rabbitwelfare.co.uk/everything-you-have-always-wanted-to-know-about-foraging-for-rabbits/
https://www.harcourt-brown.co.uk/articles/free-food-for-rabbits/foraging
http://bunnyapproved.com/foraging-101-collecting-fresh-greens-for-your-rabbit/
If your vet didn't do xrays and didn't advise on starting syringe feeds to make up for the selective eating and lack of pellet/hay eating, I'd have concerns about how knowledgeable they are with rabbits. If they aren't very experienced or you have concerns, it may be worth finding a more knowledgeable rabbit vet for a second opinion. Though I guess being a foster, the vet choice is decided by the organization you're fostering for. So maybe discuss a different vet with them if you feel a better rabbit vet is needed.
https://wabbitwiki.com/wiki/Choosing_a_rabbit_veterinarian
https://wabbitwiki.com/wiki/New_York