You describing a situation that doesn't exist. The beacon doesn't update by searching, when the app is first opened up it checks for updates. If there's a new update it downloads it. Not until the next time you connect ether the drone or the beacon to the app will the update get pushed from the phone to the beacon and it's likely you'll be forced to also update the drone before it will let you fly. You really don't want to find out there's an update when you're out somewhere and turn it on. There's only been 4 or 5 since the drone came out over a year ago so it's very unlikely. More importantly the drone will use a 5 min of so of it's battery if there's an update. Better to turn it on at home, if there's an update it gets done then, not when you're not out somewhere getting ready to fly, when that happens you'll waste allot more then the beacon's battery.
It's only acquiring a GPS signal for a min or so right after it connects to the drone. If you turn on the app it acquires acceptable GPS signal far quicker. If the signal gets weak the drone will tell you and stop flying. Keeping the beacon turned on will not help with acquisition, every time you connect the beacon with the drone it's going to start over acquiring signals. It's not like keeping it turned on gets you anything.
I know 2 other guys flying SD2s and all will go multiple flights with a single beacon charge, For all of us it's a once a day thing, in their case 3 flights or drone batteries, in mine five 20 min flights and there's some battery to spare. My beacon is over a year old with well over 75 flights so it's is far from new, every flight it get's connected to the drone and the app so there's 2 WiFi connections, not sure how much extra that uses but it's more traffic then the drone alone. As far as I know you're the first person I've seen mention the beacon's battery as a limiting issue and the drone has been out over a year. If the problem you describe is happening, you may want to talk to them about it before it gets too old, it sounds like your beacon may have a weak battery.