Before I head down this path, what are implications of fertility testing in terms of diagnostic ability and invasiveness of follow-up tests?

Good test accuracy. Fertility testing is much more accurate now than it was 20 years ago, when we did not know much about ovarian reserve testing. We used to say that 15-20% of couples had 'unexplained infertility' when all the tests came out normal, now it is way under 1% of couples who are really unexplained. However fertility tests are not perfect, couples with bad tests still conceive, some with good tests don't.
Depends on results. A full fertility workup checks the number of eggs left in your ovaries (ovarian reserve), tests for hormonal problems affecting fertility & problems with the uterus such as fibroids/ polyps. Hysterosalpingogram (hsg) test checks the fallopian tubes and uterine cavity . Semen analysis detects the 40-50% of male partners with sperm issues. The invasiveness of follow up depends on the initial results.