I go from observations farm chickens turkeys maybe ducks etc are often mainly reared in sheds, I don’t know of any confined shed reared geese, it doesn’t work. It’s not a bit of extra rain or sun that’s going to wipe us all out, it’s the human and animal viruses
It can work as a temporary measure. It did for our geese last time. As I said before, our geese spent the night in their house as usual, in the morning they were walked the short distance to our veg garden, which is fenced (usually against them but this time to keep them in). There is grass in there, perhaps not as much as they had free ranging over several acres, but enough to keep them foraging, and wild birds simply don't go into that area as the fences are high and they are scared of the small area. We feed the geese a little supplementary grain inside their house at night - as you say, gone in moments. We fill a dog paddling pool with fresh water every day for their ablutions (amazingly three geese can get in there together and still wash and preen). Our geese didn't suffer at all, I can say with confidence. The proof of our system is that we didn't get bird 'flu on our premises then. I think you have to think laterally about your problem, or cull the birds. What is not acceptable, especially as you are so close to outbreaks, is doing nothing and just crossing your fingers.
Given the way the Norfolk turkey industry works, it seems likely that the cases found in housed turkeys have been imported on lorries from the continent, so be very careful about your own biosecurity. We have large red biosecurity no entry signs on our gates, but still people just walk in and say 'oh I didn't see it'

So you have to be vigilant. You must have disinfectant foot dips with a brush at every entrance, to be used both coming and going, and be very aware of where you drive and where others have driven before they come to your land.
Increasingly I am thinking that it would be better in your case to cull your geese (as long as you know how to do so properly) and replace them when the crisis is over. You may have only four geese, but you could put other big businesses at risk, and the lives of many thousands of birds if you don't employ adequate restrictions.
I don't agree that 'it's not a bit of extra rain or sun that's going to wipe us out, it's..viruses'. Climate change is happening and if we don't do something fundamental to halt it then our Earth will simply become inhospitable to humanity. At the moment, Earth is just flexing her muscles and shaking off a few extra humans to relieve the pressure a bit (that's Earth's point of view, not mine).