This is really hard with an autistic child. I’d proceed with extreme caution on a sleep ambush. If that goes wrong it could have a whole pile of unintended consequences. I try not to erode trust unless I absolutely have to!
If she has the language can you find out what exactly she’s scared of? With my son it’s often not what you think.
My guess would be that she will need to be in control of this to get it done, and you need to dial back your worry and any pressure or this will escalate.
I’d lay out the natural consequence of this so she understands, but keep it light. Eg. The splinters aren’t a big deal but they do need to come out to keep your body healthy. I think we can probably do it ourselves but if we can’t then at 8.30 tomorrow morning we’ll go and see a nurse who does it for a job and will do it for us if we can’t manage it. Not a threat or a negotiation, just a neutral statement of what will happen and don’t get drawn into a big conversation about how they’ll do it etc.
Show her how to get it out using a cocktail stick or something in some blu tack and tweezers and then leave them available for her. She might well get curious and give it a go herself.
If she mentions pain, pop to the chemist and get a pack of numbing bandages for needles. Use one on another part of her body to show her how it will help.
All this to say, keep it light and relaxed and tell her what the options are and when you’ll need to take her to deal with it. These things are so frustrating and it’s scary when a child can’t or won’t see what has to be done to keep them safe, but those splinters will be coming out one way or another so try to relax, and she will relax.
I got my son through draining an infected toenail by watching a load of hideous TikTok videos with him about people popping spots and draining cysts, but that might be just him and not an experience I’d care to repeat!