As a matter of fact, their training could indeed have motivated their refusal to test. The Lyme test not being perfect and the disease not being that frequent in your area makes testing dubious, depending on the particular numbers. In more scientific words: if the prevalence of the disease goes down, then the positive predictive value of the test decreases as well [1][2].

[1] https://en.wikipedia.org/wiki/Positive_and_negative_predicti...

[2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2540558/pdf/bmj...