Yes they would produce more bass, you can use that and cut the budget of buying speakers.
But branded active sub at 5k is hard to get. You only option here is that you can get a amplifier separately and hook it up with the passive sub. Other options include making your own subwoofer.
You can get chip amp or such made (from ebay or from forum member) in around 1.5k or such hook it to the passive sub and enjoy quality music, or sheel out around 12k more to get a amp and sub together/ active sub.
The idea that i gave you is implemented in my car and sound great than active sub's commercially available.