Actually, I know the battery is shot in that it won't hold its charge for more than a week or 2, but it does take charge OK (in that it will, when charged, turn the bike over for a few attempts before dying). Would the conditioner help revive this battery? Worth a shot, or just buy a battery as well?
I doubt the conditioner will help, I expect it will be quite sulfated inside and this will be the cause of it's inability to perform.
I have had some success in the past with "shocking" batteries back in to life by hitting them with a couple of hundred amps from those car booster/starter jobbies (the proper ones on wheels), but the success rate has been less than 50% using that method.
Do the little (bike) CTEKs not have a desulphate phase? The one I got a little while ago for the Nissan does - it's revived the 10V battery back up to a healthy 12V, and now it just stays connected.. (FWIW the battery was new, I just accidentally flattened it by leaving the boot ajar and the light on - didn't notice for a fortnight

)
So to TBs point - yes, just leave the thing connected

Once it's finished it's condition/charge/etc cycle it'll sit there monitoring the battery, giving it a little kick if & when it needs it.