I would try the cool packs..but with this caveat:

I find that cooling off my whole body with an ice pack works everywhere. Put on the neck and wrist and it cools the blood everywhere.

They may cool your braces area from within, so try them first.
Putting the packs where the braces are may be a problem, because they tend to sweat moisture.

If moisture gets in that foam, then bacteria will grow. I had a young male patient who used other players football helmets...which had cheek foam supports. He developed a severe staph infection on his face that was difficult to get rid of (I had to compound special topical antibiotic for him).
So beware of that foam getting contaminated.