I feel your general barrier here is that you are still thinking in English and then trying to translate that thinking into Irish. The constructions that exist in English can sometimes be reflected in Irish constructions but it’s important to remember that Irish is Irish, if that makes any sense. Look for actual examples of this type of construction in Irish and copy it. I think sometimes learners try to reinvent the wheel, so to speak.

No, I fully understand that one should translate meaning and not the construction of another language.
I’m simply interested in the most precise Irish equivalent (which I know involves the verbal noun in some way) of English terms of this type.

Bean an fhir atá ag rince
“an fhear” is impossible, I think.

Perhaps you’re right.
I thought “Bean an fhir” could be mistakenly understood as a unit in “Bean an fhir ag rinnce”.