This of course isn’t FP4-specific, and you’d likely get better (or at least, certainly more constructive and less hostile) responses on more general forums, but as the error here sounded a bit unfamiliar, I thought I’d look into it a bit. Surprisingly, it is not the Google Services Framework or SafetyNet problem I expected it would be. It appears that the app is checking something new and specific about not just Google Play (the app, not the services), but about the source of the Tesco app installation. MicroG is not the problem: it appears that the app will also not run, even with a full GSF/GMS installation, if it has not been installed through Google Play itself.
I’m running LineageOS 20 (Android-13-based) on an FP4, with MindTheGapps, and Magisk with Zygisk, Universal SafetyNet Fix, Shamiko, and Props Config set to FP3. All apps I use requiring GSF, GMS, or SafetyNet are working properly, including contactless payments.
As I am not currently in the UK, I used Aurora to install the app. Running it switches contexts to Play Store, and gives an error message I’ve never seen before, of “Get this app from Play [cutesy failure message artwork] To continue using Tesco Grocery & Clubcard, get it on Google Play”. Presumably, in your case, it gives a similar message in-app. Adding the app to Magisk’s denylist does not change anything (I didn’t expect it to; it doesn’t appear to be looking for that).
In what seems like an amusing or depressing example of Tesco’s views on privacy and apps, the app starts asking for notifications permissions before it does this check, and asks every time it is started.
I can barely find anything about this error message in searches. The only reference I was able to find was for what appeared to be a paid app available through subscription through Google Play. It may be that Tesco is abusing a system meant for paid subscription apps. In any case, it looks like a new anti-AOSP technique, and it will likely take time for it to be dealt with.
With all that said, and not wanting to question your views on Google, I might point out that, if it is for privacy reasons that you don’t want Google apps on your phone, I can only assume that Tesco’s app is also extremely problematic for privacy.