Tim ...
Sorry I didn't reply earlier ... UNDER NO CIRCUMSTANCES BUY A 1200!!
I have cast this from my mind..
Ok - is this 'floatiness' what we would normally call a weave? If your bearings are notched, then the steering will be forced into one particular position by the rollers sitting in the wear notches, and it's this apparent 'stiffness' that can cause a weave because the subtle inputs
you constantly make to correct your course are efffectively negated,and the extra effort you have to make to compensate can cause an over-reaction and hey-presto we have a weave going on.
The other way that completely shagged bearings can affect the steering is by letting the steering stem wobble because it's not firmly controlled and held in position by the bearings. Again, this causes an instability in the steering because
your input isn't predictably correcting the bikes direction, more a question of it doing - within a very small parameter though - whatever it wants.. Usually this will manifest itself as a WOBBLE, or a more rapid oscillation of the handlebars.
Ok, now's the confusing bit... wobble can also be caused by tyres and some tyre profiles just wobble at certain roadspeeds (my old ST1100 on BT021s would waggle and shake at 25mph and no hands... ) and this can also be an indicator of works or knackered wheelbearings - see below...
Low tyre pressure at the front can cause a vagueness in steering at speed, and make low speed turning a b!tch.
TP 32 front 36 rear , perhaps a bit low?
Knackered or worn front wheel bearings can have
Good head bearings that are set too tight (ie that don't allow the steering to rotate freely enough) can cause a weave ...as I remember only too well when I would up the rotary steering damper on my old YR5 on a windy road just to see what it would do ... skidmarks of the non-tyre variety!
no weave felt
So ....
If the AD's checked the bearings and declared them ok, the chances are they're ok

but if it were me here's what I'd do just to check ...
Park the bike on the centre stand, then enlist the held of a large friend to sit on the bike and hold the back of the bike down and the steering straight ahead.. Kneel in front of the bike with the wheel between (ish) your knees and hold the bottom of each forkleg. Try to pull them backwards and forwards - any misadjustment of the bearing will be revealed with a clunk that you may be able to hear and / or feel as the steering stems rocks between the cups of the bearing. This is bearing slack and can be adjusted out. However, tightening the bearing may well reveal the next problem - see below!
My son acting as large friend, only just lifted wheel off ground, ok no movement backwards or forward.
The next check to make is to ask your able and fat assistant to let go of the 'bars but to keep leaning back. The bars should rotate easily and freely throughout their travel. If the bearings are notched, you'll feel it as the bearings roll over the indents in the cups - oddly in a very notchy way! You'll also probably find that the bearings will have a definite notch in the straight ahead position - remember the weave thing I mentioned a bit earlier...? If you feel this then the head bearings are sh@gged and need replacing.
Bars roated freely, no notchs felt
Now for the wheel. Get Fatso to firmly apply full lock while the wheel's still off the ground. Sit yourself alongside the wheel (you'll know which side to be on...) and give it a little spin. You're listening for any grinding noise other than the little "shussh shussh" of the discs (again you'll know it if you hear it) or perhaps even a rumbling. You can even put your finger on the spindle to see if you can feel any rumble/vibration. If you can hear/feel anything then the bearings are rough and are well on their way out.
Only the "shussh shussh" of the discs
Now hold the wheel like a steering wheel - at 2:45 or a quarter to three (you choose!) Now push one side and pull the other, do it again at 11:30

- this will reveal any lateral wear in the bearing and if you find it ... the bearing is f*cked ...
no movement here
I think there's a 'how-to' for bearing replacement somewhere...
Useful?
G