Look, if the director is going to fight you on this, then you need to push back.
Give him a ten minute warning, 5 minute warning, stand up 60 seconds before the break, and then say "excuse me, but it's time for a break - everyone, take ten, we will come back here."
Listen, if you ever get to the AEA world, there is financial burden for not taking breaks on time. And no General Manager in the world is going to want to pay overtime because the SM can't figure out how to get the director to take the break.
There is no right way for a SM to get a director to take a break - it all depends on your management style and your relationship to the director. You need to figure it out on your own. I could tell you all the tricks and secrets I have used in the past - but they were based on my management style and how I worked with that specific director - and you attempting to try to duplicate my style and method could backfire.
Look, stage management is not for the weak or fearful - you have to be a leader, and not just to the crew, to the cast, but for the entire production - including leading the director. It's something that can't be taught - it's something that you either have or don't have - I suspect you have it in you, you need to stretch your management muscles and push your style a bit.