Dyed my maroon entryway with Rit Dye and a spray bottle. Dye was mixed a bit on the strong side with very hot water. The results were an exact match and no one, when asked, can tell it was re-dyed.
Did the job 1 1/2 wears ago and still looks new. The carpet is a synthetic plush. Rit recommends rinsing after the dye is applied. I saw no need to rinse. Just a thorough vacuuming after a couple of days. There has been no color transfer to anything else even with the kittens running around.
Do be sure sure to MASK OFF any areas that you do not desire to dye, including wood.
FWIW
John