You will need to pay special attention to the amp's user manual because the 4 ohm load created by the Ninkas could present a problem if you intend to drive both sets of speakers simultaneously.
I have a NAD C375BEE which is a very capable integrated that has the A/B speaker settings. Even if purchased new, it is within your budget. NAD, however, does not recommend using both speaker settings simultaneously UNLESS ALL speakers being driven are 8 ohms or higher.
I can't speak for other manufacturers, but conceptually, it would make sense that a single amp capable of driving four separate 4 ohm loads would have to be a beast.
I'm interested to learn what else is recommended.