If you want the screws out, I'd suggest a left-handed drill bit of the appropriate diameter. And for the skeptics, they are real: https://smile.amazon.com/Comoware-Drill-Titanium-Nitride-Coating/dp/B07QKS4Z71/ref=sr\_1\_2\_sspa?keywords=left+handed+drill+bits&qid=1663466260&sprefix=left+handed+dri%2Caps%2C100&sr=8-2-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaW...
That's going to be a pain because of that low profile flat head. Here's a LH twist drill set that has a 1/16 LH: COMOWARE Left Hand Drill Bit Set, 13 Piece | M2 HSS with Titanium Nitride Coating | SAE (1/16" - 1/4") https://www.amazon.com/dp/B07QKS4Z71/ref=cm_sw_r_apan_glt_i_28KST7HR43NAP23TK87Z?_encoding=UTF8&psc=1
It's hard to tell the thread spec from the pic, but I think 1/16 LH would give you a good shot at a clean extraction. It's possible that the head will break off, but you can disassemble your cooler and twist the threaded shank out with pliers.
Ordered this left hand drill bit set (reverse bits) and it pulled the stripped hex bolts right out. Note: 2mm bit did nothing. The 4mm bit grabbed onto the bolt and ZIP, out it came.
https://www.amazon.com/dp/B07QKS4Z71/ref=cm_sw_r_cp_apa_fabc_kJV1FbXZ2EY1B?_encoding=UTF8&psc=1