Solution by Technophobe01 didn't really work for me.
Problem (just in case):
Mouse works only once. After I put macbook into sleep/disconnect mouse/disable bluetooth and reconnect again bluetooth shows mouse as connected, but mouse itself won't work (cursor doesn't move and clicks not work)
I tried to remove files as Technoprobe01 suggested - no luck. Also removed com.apple.Bluetooth.319BDC24-815F-5B78-B23E-0D73E0307B90.plist (probably yours will have different ID in the name). Didn't help either.
What helped. Remove G603 from bluetooth pairings.
Press connect button on mouse for 3 seconds - enters in pairing mode.
Find bluetooth devices on mac. Notice there might be two (2) G603 in search. Connect to the second one.
...
PROFIT. All works.
Understanding how it works.
I've tried to connect this mouse to other devices at home and noticed that Bluetooth Adress for it changes slightly for different connections: Initial mac (when it did work only after reconnect) - f1-xx-xx-98-ce-26
Android smartphone - f1-xx-xx-98-ce-28
So when I connected to the second available G603 from my mac it was f1-xx-xx-98-ce-29. I guess, the first one would have been ce-26, but I don't know.
P.S.: to check connected devices Bluetooth Adress' click on "Bluetooth" icon in menu bar while holding "Option" key.