Impediance mismatch and amplifier question
This question is not easy to answer, but I will try to give you some directions in which you can look for more info and ideas:

In audio applications one generally want the output impedance of the amplifier to be as close to 0 ohms as possible in order to ensure a "constant voltage source" (please do not misunderstand this term, this has a specific meaning - google it). This means that a given voltage output from the amplifier will generate a current through the speaker based on the speaker's impepedance (ohms law etc.). If your output impedance, as you say, is 10 ohms you are starting leave the constant voltage regime closing that of the constant current. Actually you are somewhere in-between. For a 4 Ohms speaker you will face design challenges. For 150 ohms speaker (your headset) you might be OK. I don't know, I am only assuming. The biggest assumption here is the current requirement of the 150 ohms headset speaker. Measure the output voltage from your bluetooth to see how much gain you need.
If you put an opamp between the bluetooth adapter and the headset speaker you can take input impedance as infinite, so measuring the bluetooth O/C is OK. You would need a DC-DC converter of some kind to boost the voltage supply to the opamp. You could consider e.g. LM2735 from TI for your DC conversion, and AD8597 from Analog Devices. You'll need some external components as well, these are given in the data sheets and application notes. Take a look at http://www.pcb-pool.com/ppuk/index.html if you don't make PCBs yourself.

Alternatively, you can spend time on the internet to find DC converters that doesn't need external components other than resistors. That would probably serve you better I believe.