So I'd paint the stock too, but either way I'd go to the local paint store and buy an oil based paint.
My trade gun is painted with oil based paint. It has proven to be a very weather resistant coating. Don't do Rusoleum, though, get a commercial exterior oil based paint, and use that. I tested Rustoleum and it seemed too brittle and much more easily chipped.
Sorry the focus on my gun photo is messed up...,
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Here are some other repros.
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Here's one without stain, just varnish and a painted pattern...
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LD