Are terms like "policeman" still gender-exclusive if they refer to one specific man?

Yes it is gender-exclusive, but that need not always lead us to recoil in utter horror. “Stewards must always offer any pregnant passenger assistance with her luggage” is also gender-exclusive, but I will personally undertake to assuage the hurt feelings of any pregnant male passengers who feel affronted by the wording of this guideline.

Policeman is quite common in reference to male police officers, and I do not think I would even notice it. For that matter, the corresponding form policewoman is not uncommon. For example, here are some recent uses in The New York Times:

Staring up at the steep and lifeless brown ridges looming over the valley, Gulzar Ahmad Dar, a boyish and chatty Kashmiri policeman, sighed at having drawn a short straw. [link]

As soon as my passport was stamped, a policewoman strode over, said she had a text message for me and, though I laughed at her scam, requested a tip. [link]

[…] a woman from Canada who says a handful of French policemen raped her in Paris in April. [link]

[…] he said that Muslims who were responsible for vandalizing police vehicles, damaging public property and attacking policewomen during a rally in 2011 “will not be spared”. [link]

As to whether this is “still sexist” — that seems very subjective to me, and depends greatly on the political and cultural context. You will need to decide for yourself how you feel about these usages.