Perhaps you need a stipple brush. These are most often used for stenciling, but are also used for other paint techniques. A stipple brush applies paint in an up and down motion which might be better at getting paint into the holes. Here is a picture of two types. The one on the right is typical of stencil brushes while the one on the left is most often used on walls. I do not know that this will fix your problem, but it's the first thing that came to mind.

You might be able to thin the kilz with water. Thinner paint is more likely to flow into the tiny holes. Also try using a sponge to apply the thinned paint.

With such a small area to paint, get quotes from pros who can "spray it". They use airless commercial sprayers to apply paint. Typically this could be done in one day for both the primer and paint. It will also be less thick than what you will brush on. This will allow more texture from the brick to shine through.

Are you using the right roller, it has to be for rough surfaces and loaded well with paint .

Additionally, painting the grout ( mine was inset some) was made easier by buying a pack of “chip brushes” , short wide brushes from Lowe’s , basic cheap throwaway brushes, several to a pack.

Thanks, everyone! There is no extra money for the pros, unfortunately, but I'd go that route if I could. I do have a rough surface brush and a small roller. I'm definitely going to try the brilliant spackle idea on top of the primer. Wish me luck - putting the paint on tomorrow!

Make sure your roller is for rough surfaces.