Or, would the brownish stain would eventually bleed thru any water
based Kilz, even though the stain is dry now ?


That's what primer does - cover stains. It is irrelevant what the base of
the primer is.

I've had nicotine bleed through several coats of the water-based Kilz,
and this was on a wall that had been cleaned pretty thoroughly with TSP.
I had high hopes for the water-based, but no luck. One coat of the
oil-based did the trick. So the type of primer most definitely makes a
difference.

That's nicotine, though. I don't know what caused your stain, but it
sounds to me like it would make sense to give the water-based a try &
see if it gets the job done. It's more expensive, though. Maybe you
could find it in quart size. (Our local Sears had gallons only, if I
recall).