These days, I pretty much don't install apps on my phone. I just use it as an everywhere web browser and live with the web version of various things.
My 'SLA' for facebook messages is about 24 hours, and people who send me things there learn that over time. I generally just tell them why I don't see their messages right away.
In general, I make a point of not being too responsive with any online/async notification systems, except for SMS. It's one way I keep myself from being so scattered.
You could actually do that through the Facebook.com web interface much longer than you could through the FB mobile app. Of course, you can still do it on the web, though it's a separate domain (just as, on mobile, it's a separate app.)
But since they really want you to use the Messenger mobile app, to use the web interface on mobile you have to use "Request desktop site".