I’ve seen this with Brother printers a few years ago. A firmware update fixed it. So the first thing I would do is update the printers’ firmware to the latest revisions.
If it is possible to connect your printer to the LAN via Ethernet, you can use that for AirPrint. In my home, the printer is in the same room as my router so I connected it to an Ethernet switch (along with everything else in the room). My mobile devices can still see it via AirPrint because my Wi-Fi router forwards all of its multicast traffic to the Ethernet segment of the network.
I have seen Bonjour flakiness over Wi-Fi years ago on an old Actiontec router (for a FiOS network). It would only forward multicast traffic in one direction (from the wired to the wireless network) and not the other way. So my wired devices could see Bonjour services on the Wi-Fi network, but wireless devices could not see Bonjour services on the wired network.
I ended up working around the bug by disabling the Wi-Fi on the router altogether and used an old Linksys router (in bridge mode) as a dumb Wi-Fi access point. This had no problem forwarding multicast packets in both directions.
But your symptoms (device working fine until it goes to sleep and then disappears) doesn’t sound like a network problem to me. It sounds like a problem with the printer itself, hence my suggestion to install firmware updates.