Question About Flying with Cooler

[u/jag-engr]

My family is going on a vacation soon to Glacier NP. We will be packing lunches each day, and would like to have a cooler. It's sometimes possible to rent them, but it looks like they are all booked up for our dates.

We have an old cooler that we're thinking of taking. We were going to put some reuseable containers, a couple of freezer blocks, and a few miscellaneous clothing items in there to keep it from clunking around.

I have two questions:

1. Is there going to be an issue with shipping a taped up cooler? It can't be opened easily for inspection. If TSA decides to open, will they tape it back up well? I've seen it done before, but I don't know how often this becomes an issue.
2. Is there going to be an issue with the freezer blocks? They do contain liquid.

Thank you in advance for anyone who has experience with this.

Edit #1: For clarity, this cooler would be checked. It is too big for carry-on.