Shelter wants to euthanize my socialisation case cat but I disagree

Hi all - hoping for some advice as I'm feeling heartbroken and I want to do what I can while not overstepping.

I've been fostering a socialisation case for the past 5 weeks - I actually trapped her in my garden and handed her in, and she was deemed feral but they still contacted me to foster her (I've fostered with them many times before).

She is 1 ish years old, so came with a guarded prognosis for socialisation. However, in the past 2 weeks (once she got settled) she's made amazing progress!

She will leave her room and join whoever's in the common area - playing with, and grooming and sleeping around whoevers in the room. She often sleeps near /hangs out with me while I work from home (rather than in her room). She constantly talks to us, meowing back and forth, and eagerly plays, often coming close. She's not happy at all to be touched or approached, but we can walk right by her comfortably. She's started taking food from my hand. This has all happened in the last 2 weeks, so I'm quite optimistic she's going to keep getting better. She is honestly a very happy cat coexisting, and has begun engaging on her own.

However, I got a call from the organisation today saying that they want to euthanize her as she's not making progress fast enough, they think she's not happy around humans and that she'd be dangerous to rehome. To be fair, her last vet appointment she was super duper unhappy, hissy and scratchy. However, at home she's chill and honestly a lot more comfortable than many other socialisation cases I've taken on, albeit slower to progress.

I honestly don't know what I can do without overstepping. I'm quite confident that she's happy where she is, so I don't see the harm in giving her more time and work, which I'm happy to take on. They want her euthanized in the next day or 2 so she doesn't stay stressed (she's honestly not).

Are they in the right or am I?

Any advice welcome on what I can say to them as I'm heartbroken over this 😭 I tried to gently push back over the phone, but they were insistent. I'm also slightly peeved that I wasn't included in this decision and that the caller kept referring to her as a "he". I don't want to ruin the relationship with the shelter as otherwise they've been good, but this feels wrong.

Note: I unfortunately cannot adopt her, no matter how much I want to. Hence the fostering.

Edit to add: All of your opinions, whether they be that she will improve or she won't and won't be able to be rehomed are super useful, keep them coming! I just want to know either way that it was the right decision.

And, while TNR or working cat programmes would be fantastic, and honestly I'd love that for her, it's not realistic sadly. I'm in NZ so wild/semi-wild cats are a big no for the sake of wildlife. For those who don't know, our wildlife evolved without ground mammal predators so are very easy targets!

UPDATE: I'm so sorry all, it's not a positive update. First of all, thank you so much everyone who commented. I read them all, sorry I couldn't respond to everyone. Your comments really helped and will help me for the next sweet cat who comes along.

For the update, I went in and talked to them but I had to take cat in at the same time. There was no changing their decision. I know that euthanasia is never the worst outcome for an animal, but it is still sad. They did take time to listen to me explaining her progress, but their decision was based on her lack of letting us touch her still and putting up boundaries with her space, meaning that (in their eyes) she'll never 100% be comfortable with people in her space and will always have that as a stressor. I still believe she just needed more time. I still think this rescue does great work with many animals, but I'm going to be fostering for another rescue from now on as I don't want to be in this situation again. In this whole mess I had called a couple of other rescues for advice so I have some others I can go to.

Thank you all, I got a lot of great advice from you and I'm sorry it's not a better outcome. At least she had a nice warm home, got a full tummy and knew love for a while <3