Thankfully, during COVID, mine were shipped directly to me. And also thankfully, I insisted on the max dosage after reading a study that was done on Xolair that said 80% of people respond better to larger doses. So I am at 600mg a month. On occasion, I'll miss a couple of injections so therefore I have extra. I use those during the Jan/Feb outbreak and just load up. It still does nothing with 900-1200mg a month. See if they will allow you to inject yourself. It would save time and money (if you don't mind that sort of thing). I've never had trouble with doing it myself. UPS delivers to my door in a very well chilled package and I just put it in the fridge. You just have to ship back the box - they give you a label - drop it off at UPS store and you're good. Genentec is a little difficult to order but it's still only a few minutes a month to call and schedule. They have a thermometer with the meds that will say "OK" so you know it's been chilled properly. I like it, very convenient.
I looked into Dupixent - it seems like Xolair in many ways, so out of exhaustion on all of this, I didn't pursue it. Like you, any heat build up aggravates it terribly. Sitting or laying in bed even - anything that builds up heat will trigger a worse reaction. Plus the time of day - 3p and 7p I start really itching. I have the cream - Triminiclone (sp?) but it doesn't do much either. Better than nothing I suppose.

Hang in there!