AM: morning, PM: afternoon/evening
[0] [12:00 PM, 12:00 AM) for after the noon instant
[1] [12:00 AM, 12:00 PM) for before the noon instant
24-hour time is superior to the mod-12 time system in almost every way but mechanical clock faces and soldier shifts under early systems that used sun dials. However, the equivalent question can be asked by a day is a clopen interval of [00:00 current day, 00:00 next day)
A place open 24 hours/day is open in the period 0-24.
I think you get the essential utility you want in a 24 hour clock by simply letting 24 (or "24:00:00" when fully written out) be a way to represent the end of the day.
It's also already in use, with examples at https://ioi2023.hu/faq/ , http://www.garageferrarimilazzo.com/en/ , and https://jozsefvaros.hu/english/news-in-the-district/2022/12/... .