When I had to touch up a dry damp stain, I used Dulux Damp Seal primer and undercoat. It is water-based. It is suitable for dry and wet stains.
I used it because, like you, I was worried an oil-based primer would show through the emulsion.
I used two coats of this quick drying product, and the stain didn't bleed through, and there was no evidence of this primer showing through the emulsion.