Strategy without timing is just a good idea at the wrong moment.

The same request can be obviously reasonable or obviously annoying depending on when it arrives. Content is only half of it, and the other half is the state of the person receiving it. An idea delivered at the wrong moment does not get rejected on its merits, because its merits never get considered.

With your next ask, spend as long choosing the moment as you spent on the argument. Not during the rush, not as they are leaving, not in the middle of something going wrong. A good idea will wait a day, and a day is usually all the difference.