The best way to paint over chrome is to give it a light sand blast , epoxy primer then color coats. Adhesion promotors will work well too on CLEAN and PROPERLY prepped chrome. I would not use self etching over and adhesion promotor. To many single component sub straits will cause it to chip easier. Epoxy or even a 2k urethane primer/sealer would be all you need before color coats when using the promotors.
As for heat effecting the paint on the grille. I call Bullchit. It will not be effected IF it's prepped right.
This is a 14 grille shell that was chrome. Hell...even the center section is tinted and cleared over with just a promotor. No issues and it's got over 20k for miles out here In the desert and it's seen some HOT azz temps too.