Typical silk screen "engineering" is performed in such a manner as to provide very high resolution and high fidelity to the original artwork. For the project you describe, dealing with three dimensional objects, you are not going to achieve such fidelity, nor would I expect your objective includes that aspect.
You could use an ordinary spray paint if you do not require to later remove the material from the screen, as is commonly done with conventional silk screening. You should consider to use water based paint to ensure the solvents will not dissolve the screen.
Alternatively, one can purchase an atomizer type or pressure type spray device and use the liquid mask, again making certain that the materials are not going to react with each other.
If you use any type of spray method, flatten your items as much as possible, unless "fuzzy edges" are part of the art work. Use light coats multiple times rather than few heavy ones. You'd want to keep a constant angle from the spray to the art pieces to the screen, in order to reduce the under-spray which would result from spraying under the overhangs of the items.
You mentioned branches (sticks) in the piece. If you care to increase the level of success, consider to sand flat the portion of the branch contacting the screen.