43t34fg4 I was interested in the reply @ErnestThornhill mentioned here but he's stopped responding it seems. It could very well be what he mentioned.
They're suggesting that maybe Sam's Club is checking if the phone has an OS certified by Google on it using Play Integrity. It's very possible that that's what's happening.
The logs you shared had nothing of interest that I could find. And even if the Sam's Club app does choose to not work because of this, logs may not be of any help.
edit: I'm changing the title of this discussion because GrapheneOS isn't blocking anything. If the app still isn't working with exploit protection compatibility mode on, then the app may be doing this because of Play Integrity results.