Whenever anything "bleeds through" a primer or paint, what's actually happening is that some chemical in or on whatever you're painting over is dissolving in the paint you're using and diffusing through the wet paint film to it's surface where it discolors that surface.
MOST things that are soluble in water are not soluble in mineral spirits and vice versa, so if a stain bleeds through an oil based paint, it generally won't bleed through an emulsion paint and vice versa.
The brown spots you're seeing are probably "tannin" from the knots in the wood, and tannin is soluble in water.
I'd use an INTERIOR oil based PRIMER to paint over those spots only, and allow to dry, and then paint over that oil based primer with paint to match the rest of the woodwork.
coupla PS's:
PS #1:
One of the primary differences between INTERIOR and EXTERIOR oil based primers is that if an exterior oil based primer claims that it "breathes", the way they can make it do that is by adding enough coarsely ground white extender pigments to it so that it actually dries porous.
You see, oil based paints won't allow moisture to pass through them the way emulsion paints will. But, it's important that wood or masonary on the exterior of a building be allowed to dry out if it gets wet from humidity passing outwards through the walls. So, in order to allow water molecules to pass through an oil based exterior primer, what they do is add enough large coarse "extender pigments" to that primer so that as the solvents evaporate from the primer and the binder shrinks as a result, tiny air passageways open up between those large extender pigments so that the primer dries porous. And it's this porosity the primer manufacturer will point to to make the claim that the primer will allow any wood or masonary under it to dry out.
Interior oil based primer won't dry porous. Also, because wood outdoors expands and shrinks MORE with the larger variations in relative humidity you get outdoors, exterior oil based primers need to dry to a softer film that can stretch and shrink more without breaking (and cracking and peeling off).
PS #2:
The EASIEST way to help eliminate brush strokes in any paint is to thin it a bit. With latex paints, you thin with water. With oil based paints you thin with mineral spirits. The more you thin a paint, the more it shrinks in film thickness as the thinner evaporates from it. So, put a bit of interior oil based primer in a small container and thin it about 10 percent up to say 20% by volume with mineral spriits (also called "paint thinner") to help eliminate brush strokes. Put on a second coat if you want to compensate for the thinner dry film thickness it will dry to.
On vertical surfaces, thinning the paint can lead to other problems, notably the paint "sagging" on the surface as it dries. Some companies make "paint conditioners" which will thin oil based and emulsion paints without lowering their viscosity so that you can achieve a smoother paint job on a vertical surface using a brush. Look for "Penetrol" for oil based paints and "Floetrol" for emulsion paints in your paint store, both made by the Flood Company.