you really should try this out on scrap first as wood stain is meant to penetrate and travel into the grain of the wood.
I would think you are going to have a blurry splotch at best plus the stain will wick under the screen and make a mess.
i guess if I were going to try this I would try some cut vinyl possibly a low tack type as my stencil, attach it to the table and dab your stain on by hand let dry then pull the vinyl.
You might look into Minwax gel stains as they are not watery and are actually a gel you apply then wipe off after a while.
The gel will still penetrate the wood but you might have a fighting chance with the thicker stain.
You also might try doing this in reverse, again with vinyl except this time as a mask. Cut it as your image applty to the top then seal the wood with a light coat of the following finish......40 % polyurethane varnish + 40% tung oil (see homer formby's brand good stuff)
http://www.formbys.com/products/tung_oil/and 20% mineral spirits or paint thinner.
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Apply the above coating with a clean cotton tee shirt (white) let dry peel your mask, chase the adhesive then stain the uncoated area that was under the mask.
Go back and hit everything with the finish mix, several coats it builds up nice and is buttery smooth.
TEST TEST TEST it first all in all a long shot.
You might do way better with thin vinyl on the stained top and a heavy top coat AKA decoupage like you see on some bar tops abd restrauant tables.
https://www.google.com/#q=using+a+decoupage&spell=1mooseman
mooseman