I think to start off with just go with the active Audio Engine.
Yes later down the line you might want to split out into an amp and passive speakers.
If I was in your shoes I would want to spend more than $200 on an amp. I think trying to split out for your inital budget might give you a worse outcome.
Disclaimer: I am bias as I love my Audio Engine A5+ for my laptop/working setup. I also have separate amp/speakers in my main living area.

