I am surprised you are not able to request a fridge?  maybe not all hotels have that option.  sometimes you can request a fridge and a microwave for your room for a small extra fee. 

Natalie,
I also have to travel for work, although not as much as you do; mine is more sporadic, on an as and when needed basis, but for many days at a time. I also had apprehensions about the refrigerator but from experience, in most cases, the hotels provided one (just complementary) when asked. Even the very basic ones usually keep one or two refrigerators as spare for special needs. Tell them upfront I have special needs and will need one. If they are not open, they usually they will provide one for a fee. Also, whenever I am traveling, my preparation usually includes researching how far the nearest health food store is so I can buy the almond milk and other things from there.
When I was on list one, I use to pack heavy because I usually carried a lot of ready-to-eat things with me- olive oil potato chips, rice cakes, vegetables like carrots, celery, cucumbers, lettuce etc. I also carried brown rice cereal so that after I get the almond milk, it makes for a good breakfast. Boiling eggs in the microwave works too. For eating out, check restaurant menus in advance online and call and ask them about ingredients and their openness to make modifications in the meal for you.
One other tip, sometimes to be safe, I use to eat the boiled vegetables from the hot bar at the whole foods store, but would land up flaring. Someone there had told me they use olive oil so I thought it would be safe, but later found out, they use margarine to stir fry the vegetables and the soy in margarine caused me discomfort.
For other tips on food when traveling, do a search on this site. There was some extensive discussion a while back, including types of foods to look for, foods that would be least likely to cause flares. It has some very valuable information there.
Hope this helps!
-Hami.
 
 

Yes, search "travel"/"traveling" and many informative posts in there.  Also, Nicole suggested the Koolatron softbag cooler you can plug into any outlet or plug into your car, strap to carry on the go -- can get at Amazon, Sams Club, etc. fairly cheap.
And I also have been surprised how many places whose websites say they don't have fridges will provide one if you just ask.
Even if you can't get a fridge, if you have any sort of cooler with you each day, worst case you could get by w/ rice cakes, precut veggies like celery and sliced meat that don't require any prep and aren't messy to eat on the run.  Would just require more trips to a store.
My heart goes out to you... following this diet is tough enough when you're at home trying to cook everything.  I can't imagine traveling while dealing with this.  Despite only being on List 1, you have a lot on your 'plate.'  hah, ok, bad joke, but I promise List 1 is not forever.... even when you get to 2 and some of 3 it will be feeling so much better to get that bit of cheese or have tomatoes again... this is only temporary... you can do it!
 

you might look into http://www.monasheespringwater.com/index.htm