Bedtime, tell DP not to panic and TO BE A MAN !!
Sorry, do not mean to be flippant, but for obvious reasons a simple, non-invasive test for the man has to be performed first before tests that involve risks for the woman are to be considered (like laparoscopy/hysteroscpy, both camera tests of your pelvic organs).
If your GP has not given you a little pot and instructions how to get the sample to hospital (needs to get to lab as fresh as possible, usually within 2 hrs of production) he may need to do it in hospital. You can assure him that he will get privacy and time.
Most men do not relish the thought of having to give this particular type of sample, but honestly, my sympathy has limits as beyond the initial blood tests women have the more uncomfortable/riskier/more time consuming tests - and all going well, we are the ones who "enjoy" pregnancy, childbirth and usually the bulk of child care.
So: see above. Stiff upper lip (and something else ..) and he will be fine. And hopefully result will be too. BTW, the fact he has one testicle does not matter as the test looks for No of sperm/mililiter, usually @ 20 million/ml.