All of this assumes that the G413 is even capable of receiving firmware updates. Some of the (usually higher-end) mass-produced keyboards out there are update-able, but many are not...the keyboard I talked about in my initial response that had similar symptoms with a dead controller had a chip on the main controller that was only "OTP" (one-time programmable); that is to say, the PROM on the microcontroller could only be written to once, at the factory, and that was that: it could never be changed again. Despite this, the chip acted like it had no ROM contents at all...how it's possible that the ROM was wiped on an OTP chip, dunno. Maybe it took a static discharge or something that fried it. All I know is that in the end, whatever the cause, it made that particular controller a complete door-stop. Only solution was wholesale controller replacement (or to throw keyboard in trash and acquire anew).
Anybody who wanted to help with your particular circumstance would have to know a lot more about the architecture of the G413: most importantly, what microcontroller is it based around. Until we know that important detail, how to put your particular keyboard's microcontroller into a mode where it can receive new programming (assuming that it even can) is a complete question-mark. It is also entirely possible (and even probable) that if the micro in question has any sort of flash protection feature (e.g., requiring validation of cryptographic code-signing by Logitech for it to accept a flash image), Logitech is probably using that to assure that only they can push firmware images to their keyboards.
In short, there are so many different variables that can go into this, that unless this is a keyboard that has a very strong & broad community and user-base around it (one that has done the spadework to tear it down, document its architecture, has reverse-engineered the ROM flashing protocol & exploited any vulnerabilities that might be needed to sideload unauthorized code to it, etc.), you aren't going to be able to resurrect this device yourself without a LOT of hard work and very long hours. So then it becomes a question of diminishing returns and whether it's actually worth it. The most sure path to success would be to talk to Logitech and ask if they can repair keyboards like yours out-of-warranty, and then if so, determine whether the cost of such a repair is even worth it compared to buying a new 'board.
There are a billion different keyboard models out there, they're all different internally, and so sadly there is no one-size-fits-all guide on how to "re-flash a keyboard's internal controller".
