BPA is in #7 plastic (confirmed in the article posted). Plastic coolers are #2 plastic.
My Igloo cooler is HDPE (#2) only on the outside. The liner is polypropylene (#5), another one of the plastics that are considered food-safe and heat stable, though there may be reasons to avoid going over 160-170F. PP doesn't contain BPA, but it can release trace amounts of quaternary ammonium, which might have reproductive effects but is also a pretty ubiquitous anti-microbial used in food preparation, water treatment and cosmetics.
I assumed all coolers were lined with the same plastic.