Logitech Anywhere Mouse - even new batteries show 'need to replace' red blinking light

I've had this same issue. The 3 x orange 1 x red flashes is an error code rather than low battery.

I've managed to get my mouse to work while holding the right mouse button down for at least one second while switching it on.

I've just held down on the right mouse button (and a few others) while powering it on and nothing worked.

I've tried various combinations but it was when I decided to keep holding right mouse buttons and both small thumb buttons on the side and held for a longer time when I got a brief yellow light after about 15 sec. After that, I was able to power on with a green light and all is well.

There seem to be different RESET scenarios with different button combinations. Just try all of them and remember to keep holding down for longer than 10-15sec. That's the part I've never read anywhere else so far. Good luck!