Re: gsxr 600 not charging the battery??
There is a similar thread/s on the go at the moment.
From what you are saying: your battery charging rate on the bike is too low, the battery is discharging as the bike runs Ie nothing going back into it.
The ECU is probably shutting down the steering damper to conserve power or there is not enough power to operate it.
You need to investigate further.
The uk Suzuki site lets you input bike reg and vin number to check if any recalls are due/overdue, WARRANTY section of the site. Dont know if the U.S. site has this.