I worry about bottle cages failing by fatigue anyway; there are very few that appear to be able to withstand, say, and actual bottle being in them for very long..... and that is even though the loads are mostly in a central plane. Once mounted on the side of a fork at a jaunty angle I'd expect many such fittings to be a (fairly short lived) fatigue experiment.
Regarding stability and loading; having mass attached to the fork ups the moment of inertia (MOI) in the steered part of the bike. This does make the bike handle differently but you soon 'get used to it'. The one thing that may happen with an uneven load as you intend is that the bike could handle a bit weirdly over bumps; when viewed in the reference frame of the whole bike, the steered part doesn't always rotate around the central point; because the bike can be tipping to one side at the same time and accelerating (e.g. in exaggerated form if -say- the front wheel is going into a large pothole off-centre). Because the tyre is now on a camber the contact point changes and the steered part of the bike sees a lateral thrust in addition to the other loads that arise. The net effect of all this is that (instantaneously) the steered part of the bike may be turning around an axis that is offset sideways vs the steerer centreline.
Because the MOI is sensitive to this in the proposed arrangement you may find that the net effect is that the steering kicks faster to one side over bumps than the other. You may not notice this or you may just 'get used to it' (there's quite a lot of that going on, on most bikes...). Note that if the view down the steerer changes between hands on and hands off, the bike isn't tracking perfectly (usually because of a frame geometry issue, sometimes uneven loading -which can be the rider-). In this event every bump (even a head-on one) is liable to give a little impetus to the steered assembly and this is exactly the sort of thing that can provoke a shimmy.
FWIW when assessing a new loading scheme/bike, it isn't a bad idea to ride no hands (or as close to it as you can) and then to deliberately perturb the steering by knocking the handlebar. Try both directions, if you might have an uneven load or the bike might not be tracking perfectly for some reason. Also try this at different speeds.
You can mentally mark the whole scheme down if
a) the bike can't be ridden no hands, either easily or at all
b) the bike behaves very differently when perturbed in different directions
c) the bike behaves very differently at different speeds
d) you get a 'high energy shimmy' developing over a wide range of speeds.
All bikes will shimmy; its just a question of at what speed and how dangerously. The energy contained within a shimmy varies, but the highest energy is when there is lots of mass that is coupled to the moving parts and not very much damping. When there is no damping, starting the shimmy can be very speed dependant but once it starts it can carry on over a wider range of speeds. Thus most unladen road bikes can be made to shimmy violently starting at one specific speed but because the main mass in the system (you) isn't well coupled the energy contained within the shimmy is usually small; in the test I have described (with a road bike) you will usually find you can start a shimmy and it can be so bad it would soon cause the frame to fail if it were allowed to continue, but can be quelled easily by simply putting a hand back on the handlebars. If you know what speed a shimmy can start at, you will know what it feels like and how (not) to hold the bike at that speed.
If you start to get a shimmy developing in a bike with a load on the steered assembly, you are in a whole different problem; by its very nature it will be a high energy situation and some major effort /change may be required to bring it under control. If you have an offset load as you describe one possible effect is that the range of speeds over which a shimmy might start becomes broader.
So I usually test road bikes until I get a shimmy and then see how bad it is and how difficult it is to get it back under control. But with a touring/loaded bike I tend to make sure that in 'normal riding' I don't get a shimmy and then in use I stick within the range of speeds with which I am happy. I have seen one or two loaded bikes start to shimmy and it was ugly; pretty dangerous in fact. However I have seen many more instances where the load simply wasn't attached well enough, and just started to come off, often because something broke or came loose.
The vast majority of bolt-on schemes for attaching loads to forks seem woefully under-engineered to me; the whole scheme ought not overload any one bolt (remember that the loads can be multiple and don't always add up in the way you expect) and ought to be proof against any one bolt/stay/clip breaking or coming loose. Most schemes are not like this at all, and many of them I would not consider fit for riding down the shops with, let alone more lengthy jaunts on rough surfaces.
cheers