The battery widget almost certainly needs to update how it parses Bluetooth metadata or update an out-of-date hardware ID list that should include all major Mac-specific bluetooth keyboards.
Bluetooth identifies it as a keyboard correctly, indicating Logitech's side of this is likely correct. This is most certainly something that should be fixed on the Apple/Widget side of the equation. I am forgetting to recharge my keyboard because only when you use the "large" widget size does it differentiate the keyboard versus the mouse.
I was genuinely surprised that Apple's biggest accessories partner's most popular Mac products aren't properly supported.

Correctly identifies MX Keys Mac keyboard and applies icon.

Incorrectly applies icon despite having different metadata pulled from same source as the bluetooth menu.

Confuses uses not using a gigantic widget for their devices. I am recharging mine right now because I mistakenly was only recharging my mouse due to this bug.