My 2006 runs as described when the battery gets weak. The way I check for this is to leave it on the tender overnight and it will run perfectly first start in the morning. Then it will get worse as the bike sits. Changing to a new battery solves the issue (until the new one starts doing the same thing, sometimes in as little as a year). I switched to a lithium battery a few years ago (just because I had one on the shelf that fit) and haven't had the issue since. My hypothesis is that the voltage starts to jump around as the battery gets weak at low revs and causes the ECU to change mixture. New batteries can really be hit and miss these days, so if you buy one, get a brand name.
I also worked on a member's bike that had similar issues (but far worse, it was almost unrideable below 3k rpm) and we found after much searching that one cam was off a tooth. However, the previous owner had had a valve adjustment done right before selling it which doesn't sound like that could be your issue.
I also worked on a member's bike that had similar issues (but far worse, it was almost unrideable below 3k rpm) and we found after much searching that one cam was off a tooth. However, the previous owner had had a valve adjustment done right before selling it which doesn't sound like that could be your issue.