If you want not shitty speakers and subwoofer, you're going to need to up your budget by a lot.

The Edifier R1280DB is within your budget and have Bluetooth, but no sub out.

The Edifier R1280T has a sub out, but no Bluetooth.

The Klipsch Promedia 2.1 is the best sounding 2.1 system under $300 IMO. No Bluetooth though. They used to have a Bluetooth model, but I don't think it worked very well.

Here's a new offering from Klipsch that looks promising. It has Bluetooth, but is a bit above your budget.

Otherwise, look up something from Kanto or Monoprice if you're willing to up your budget.