When you say "chronically light up", what do you mean? Do you mean the control panel on the printer itself?
Once a WiFi connected printer is initially connected to the network, said printer will always be connected to that network if it, the printer, is turned on.
I had to connect a new Brother monochrome laser printer for my folks, and it's the first time I've ever connected a printer via WiFi. Every time they power up their system (HughesNet and computer) their computer gives them "printer connection failure" notices if the printer isn't turned on, as well. I've searched and searched through the settings and I can't find anything, thus far, that will stop this. They use their computer nearly every day, yet print something perhaps once a month. It's stupid that they need to have the printer on all the time. 😞
I should have explained a little more thoroughly.
I was able to wirelessly connect my folks' printer without any issue. They have Gen4 and a 3rd party router. I followed the Brother directions and first installed the Brother software on the laptop, then, through the control panel on the printer itself, I was able to wirelessly connect the printer to the router and network. That all went without issue, and it works very well. They've printed quite a few things when they first got it, and have printed invoices from online shopping since then.
The problem is that, when they start the system in the morning (modem, then router, then a little later, the laptop), if they don't also turn the printer on the Brother software on the laptop complains, in the form of a fairly good size message in the middle of the screen, saying that it can't communicate with the printer, or something to that effect. So far I haven't been able to find any way to stop this from happening. There's also no way to X out of the message, and it always stays on top of anything that is open. I could teach my stepfather how to use the task manager and stop the message there each time, but with his very limited knowledge it may be a bad idea for him to be messing around with such. And, it would probably pop back up a few seconds later. So, the only thing they can do to avoid the debacle is to turn the printer on every day, even if they have no plans to use it. It's a pain.
I'm sure there is some sort of settings adjustment or option to stop the message, but I've not been able to get over to their house to spend any real time trying to figure it out. They have two dachshunds that want to kill me, or at least chew on me a little bit, so I'm not able to spend the time there that I would like to.
With all of this said, the printer does go to sleep if it hasn't been used in a few minutes. So at least it's not really "on" all day. But...still.
With regard to the issue you're having, I don't really have any good ideas as to what could be going on. Hopefully someone with more experience can help. I wonder if could be that it's some sort of auto setting that brings it out of standby when it senses a device is near (by the strength of the signal) so that it can be ready to print. I don't know if printers can do that, though, nor if it would make much sense for them to constantly be doing so.
I use an HP Envy 5530 wireless printer. (Modest price) It was effortless to use this printer with Hughes Net, Never a problem.