I wonder if a headache hat might be helpful for you. Maybe not as intense as sticking your head in ice, but it'll cool your head and you can stay upright.
Here's one I have: https://www.amazon.com/Migraine-FOMI-Headache-Wearable-Freezable/dp/B078N1PW5D/
There are many other varieties, too. I hope you find something useful! Vertigo is absolutely miserable. Sending my empathy and support. -Melissa, migraine.com team