Insulin is fine out of the fridge at room temperature for about a month.
If you take insulin out of the fridge for a week, you can return it to the fridge. Next time you take it out, it will last at room temperature for about 3 weeks.
You only need a cooler for insulin pens if they will be kept somewhere hot such as inside a car without airconditioning on a hot day. Storing it in a hotel room should be fine provided it is not kept in the window or on a radiator.
Personally, I rarely trust hotel fridges as they are rarely maintained well so may be too cold which will damage it more than keeping it at room temperature.
If you need a cooler, the most recommended is a Frio wallet. I have been using mine for 20 years when I travel somewhere very hot such as trekking in Morocco. These are not electric coolers - they are much more convenient and smaller.
I recently travelled to Dubai where the temperature was 40 degrees at midnight but I did not need any cooler for my insulin - it was fine as my bag provided sufficient insulation for that short time drinking at a roof top bar.