Finding out the answer to this issue doesn't require a "battery of tests", because the potential problem is with a man. The test requires one (1) sperm sample and one (1) minute with a microscope. It's the same test that's done after every vasectomy. The test is cheap, simple, and straightforward. Either he's making a normal number of sperm, or he's not.

Fertility screening for women is much more complicated, and carries health risks from iodine-based dyes (used to see whether the fallopian tubes are blocked) and other procedures. But a quick check on a man is really quite trivial. In fact, your average second-year biology student could perform this test with an acceptable degree of reliability.

I suspect that the man feels rejected over these requests. He may also have a 'fertility fetish', which is not unusual, and consequently believes that knowing he is "shooting blanks" would be a turn-off. But given the low physical and financial cost of the test, the woman's request doesn't seem disproportionate to me.