Ice and dry ice are prohibited in the cargo compartment except in registered freight shipments that use special packaging. When you check a hard-side cooler as baggage, the interior MUST be visually inspected by airline personnel before it will be accepted, to verify that there is no ice or dry ice inside.
That is separate from TSA inspection. If TSA sees anything suspicious when the bag is xrayed, they will cut the tape to get it open, primarily to check for possible explosives. Your best bet is to put the content in a trash bag that you tie closed after the airline employee has inspected it, and then hold down the cooler lid using a luggage strap. That can be removed and re-strapped any time anyone needs to inspect the cooler. Having the content in a trash bag means that you can easily pull it out at your destination in order to get your ice in.
You should not check frozen meats, fresh fruits or veggies, or liquids in glass or soft containers (like juice pouches, for example); if the cooler gets lost the perishables could rot, and glass/pouches of liquids might leak if the container is dropped. Also, frozen meats (or any kind, really, such as a cured salami) are very dense, and will trigger suspicion on the part of TSA.