NGram

What you’re describing sounds a lot like eosinophilic esophagitis which indeed generally cannot be tested with the skin prick test (though the skin prick test can sometimes work, lack of allergens found in a skin prick test does not guarantee that your esophagus will not react to those allergens).

The best option is changing your diet to experimentally narrow down allergens. You can do it somewhat like a binary search though I don’t think medical professionals will recommend that since it can lead to malnutrition. I’ve been recommended to sequentially eliminate common allergens by doctors, which is a safer option.

To address the original question, blood tests for allergens (assuming they work) would have the same limitations as a skin prick test (the flaw of any general test for a localised problem). Though I suspect a blood test would work a lot less well because it’s no longer connected to your immune system, which is responsible for allergy response.

I’ve mirrored it to my own git server too https://git.ngram.ca/mirrors/cdrm-project I will ignore DCMAs because I (and the server) don’t live in the USA.