When a stain bleeds through paint or primer, what's actually happening is that the stain is dissolving in either the water or mineral spirits of the water based or oil based primer or paint.
KILZ is actually an ordinary oil based primer. The difference is that instead of using 100% mineral spirits as the thinner like other oil based primers, it uses a 60% naptha and 40% mineral spirits mixture as the thinner. Naptha evaporates many times faster than mineral spirits, and so KILZ dries much faster than an ordinary alkyd primer. The whole idea here is to have the KILZ become too thick too quickly for the stain to migrate to the surface and discolour that surface. That would result in the stain "bleeding through" the KILZ.
If the stain was caused by water, or anything water based, it probably won't dissolve in mineral spirits, and you could probably use ANY oil based primer to hide it.
Also, I see absolutely no reason for repainting the whole ceiling with KILZ. If the KILZ area shows through your paint, you just need to apply another coat of paint.
About ceiling paint. Ceiling paint is nothing more than a low quality wall paint. They don't use as hard binders in ceiling paints because you generally don't have to scrub stubborn marks off of ceilings like you do walls. If it were me, I would use a higher quality wall paint on your ceiling to get better hide of the white spot caused by the KILZ (or other primer).
If you do opt for using any other oil based primer, try to get a high hiding primer. Often primers don't have good hide, and that requires several costs of primer or paint to cover a patch.