I'm not familar with working with acrylics or heard of mixing dish soap with them, but hey...
I am an old hand at doing car grilles!
I use enamel (Testors or MM) flat preferred or sometimes semi-gloss black paint.
I wash down the item with a Q-tip and Isopropyl Alcohol (% doesn't
matter) to remove any oils. Once dry I apply enough black paint so that
it flows down into the recesses. I leave the part laying painted side
up until dry (usually a day or two). I then rub the high spots with a
flat toothpick using the wide rounded end to bring back the chrome
highlights. If you rub gentle enough the wooden toothpick doesn't harm
the chrome plating. I even trim or sand the end of the toothpick to
whatever shape I need to get into tricky corners. It's the safest way I
found to save paint (no mixing of washes) and keep the background from
disappearing. Since those areas aren't handled really, it doesn't wear
off.
Just my old school way of doing it.
Good luck.