It seems that what you need to do is to mix something with the epoxy so it doesn't run. The trouble is that just about every thixotropic agent that I'm aware of does add some sort of tint to it.
I think you should do a little experimenting before tackling your masterpiece!
I would try mixing a 3 ounce batch of epoxy with some cab-o-sil...just enough to thicken it up so it won't run. It wouldn't take much. And don't forget that the thinner the coat when you apply it to the wood, the less it will be inclined to run anyway.
When you're adding the silica thickener (cab-o-sil) the epoxy will start to whiten up, but when you spread it out, the colour will nearly disappear.
So I would try a test batch on a similar piece of wood to see what happens. As long as you don't put too much thickener in the epoxy, you should still be able to paint it on your wood with a brush. I use those cheapie white brushes from Lee Valley. Clean them right away with acetone, and you can use them several times.
Something else I should mention and that is the temperature of the wood itself.
If you can warm the wood up by keeping it in a hot room for a day or so, take the masterpiece out of the hot area and then add your epoxy. As the wood cools it will draw the epoxy in to the nooks and crannies.
If you do it the other way round, that is to say epoxy the wood and then put it out into the sun for example, the wood will off gas and leave hundreds of tiny bubbles on the wood surface. Fortunately, there is a cure for that but it's better to do it the right way.
The cure is to fan the bubbles with a hot air gun and they will break almost immediately. But you need to keep watching the surface for new bubbles to appear.
OK, fess up...which one of you clowns stole my sig? It was right here a second ago.