I think OP was suggesting that if you hint (or state explicitly) that men's restrooms are unsuitable for nappy changing, you'll be able to avoid an unpleasant chore (by pushing it onto your partner).
Honestly, changing nappies isn't that bad, particularly for your own child, and even if it was, this is a pretty selfish attitude to take.
Perfectly doable pretty much everywhere if in need, dont need to go to restroom, specially if its filthy (just take a bag and moist napkins - and tell f o to everyone who complains :) ). Its like breastfeeding - nothing to be ashamed, needs to be done - if no good place provided, find it yourself.
Don't give in to that kind of societal inequality - change it.