It's hard to speculate what your partner might have without knowing more details about his rash. The timing indeed suggests he acquired it sexually from you (assuming he hasn't had other partners since you've been together), and you are right to be concerned about herpes and yeast as the 2 mostl likely possibilities. If you really had a yeast infection recently, that's likely the better bet--but be aware that the single disease most often confused with herpes in women is yeast infection. So unless your yeast infection was diagnosed by a knowledgeable health care provider--and not by yourself based on symptoms alone--I can't be confident that's what you had. Herpes indeed can be asymptomatic. At least 25% of the infection has genital HSV-2 infection, and most don't know it.
It is correct that a herpes lesion must be present to test directly for the virus. But any doctor in the US, including yours, can send blood to a lab for an accurate herpes blood test; see numerous other threads on this forum and the FAQs about herpes testing.
So, bottom line: I suggest that you and your partner go simultaneously to a provider who is familiar with STDs and genital infections, such as your local health department STD clinic. That's the best way to know for sure what both of you have and what infection(s) you might be sharing.
Good luck-- HHH, MD