the thing to consider is that this was probably originally treated by pressure treatment. It is unlikely you will ever get a treatment to last as long as the original without removing and soaking in a bath of treament!
I've successfully used a garden sprayer to apply wood treaments but they must be water based to be thin enough to go through a pump action spray. Also it will require cleaning every few minutes to avoid blocking the spray tip and shake the pot to avoid the treatment settling out. Ideally use an old sprayer as you won't be able to use it for anything else after. Cover the ground underneath and spray on a still day to avoid painting everything in sight!
As the screen isn't solid some spray may get through to the other side.
Another method may be to use a small, 4 inch, decorators roller. slow but it does work. This creates far less mess but you still need to place a cover over any the ground below to catch the drips. I use them whenever possible now. always wear gloves and a mask.
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as this is bamboo and reed it was probably never treated from new.
Also you may have a problem with the grey/black rot on the surface.
I wouldn't suggest wire brushing to remove this and to prepare the surface as the types are easily damaged.
Instead I would spray with a very light colour treatment and repeat a couple of times to get a colour over the grey.
Only ever treat dry material.
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