Okay, so no ice. How about cold water? Our water from the tap here gets pretty cold this time of the year. I know yours probably doesn't but maybe some iced down water so it isn't 'icy' but cold so you can hold your hand in it to the count of maybe 10. Make compresses out of that and gently apply to your neck/face. If it feels too cold then let it warm up a bit and try again.What about swelling. . They said ice but it is too cold to apply ice. The swelling isn’t getting better. I can call them on Monday.
Hopefully even just coldish water would help bring some of the swelling down. Especially if you had a thick towel between it and your skin. The cold water circulator they sent home with me after my shoulder surgery you could make real cold or mildly cold by how much ice or frozen water bottles you put in the canister. It had a towel between it and the dressing on my shoulder which was about an inch and a half thick. After they pulled the sutures I had a towel between the pad and my shoulder. It never got colder than I could tolerate.
I can see their concern about freezing or frost biting your skin on your neck or face. It's very thin there but even a cold water bottle out of the fridge might give you some relief.
That way you wouldn't be dealing with ice but with cold water. Personally, I'd try it and see how it felt. Make sure to wrap it in a wash cloth or soft cloth and not apply it for more than say 10 minutes out of every hour.
Have a great sabbath and see you tomorrow after sunset. Enjoy the nice weather we are having. We are.
She's going to put me in the hospital or the morgue, lol. 
