In reference to the booze issue, unless I’m really missing some type of unknown 4th amendment issue, if the cooler is closed, and there is no sign of booze visible, the only way to look inside it would be a search warrant. Just sayn’
While 4th amendment protections generally do protect us against unlawful searches per se, there are exceptions to the search warrant requirement.
I'm not familiar with the specific laws, rules or ordinances in that area or TN, so be sure to check for any type of implied or informed consent laws or rules, posted or otherwise (e.g., vehicles or containers on these trails/property are subject to search).
Also, the same exceptions to a search warrant for an automobile or vessel would apply to a SXS (e.g., consent, exigent circumstances, hot pursuit, plain view/feel, incident to arrest, etc.).