My feeling is that the actual run time is right now longer than 10min, but you can't tell because the light dims down into the IR, so you don't see it. When you lower the voltage on a halogen lamp, it gets dimmer, less efficient, and more red. And you are operating at a much lower voltage than the lamp is designed for. Of course you might also have a problem with the battery; you will see poor battery run time is a single cell in the battery is defective.
Going at 12V, you would expect the run time to be shorter, since more current would be flowing. However you will get much more light out of the
bulb more efficiently, and the lamp will run at full brightness for most of the discharge period.
I believe that if you go to 12V, and use a 12V, 20W bulb, then you will get a much more useable light for a greater run time.
Please take a look at the LVR link that I sent. Building a regulated system is actually quite simple. You essentially splice the LVR in series with the lamp, and with little else you will have a functioning _regulated_ lighting system. Even if you don't want a regulated system, the page describes all of the issues associated with changing the voltage applied to a halogen lamp.
With regards to run time, even at 12V, you are not going to get a very long run time. The current required by the lamp is 4A (4000mA), and your batteries are only 2Ah (2000mAh). This means that _at best_ you can only expect 2 A h / 4A = 1/2 hour of run time. If you want a longer run time you will need higher capacity batteries (meaning more Ah) or lower power lamps (meaning fewer Watts).
As you noted, the output with the current system is acceptable to you...but you are using the lamp in a very inefficient fashion. Running a 20W lamp properly will produce more light and consume less power than your grossly underdriven 50W lamp.
Regards,
Jonathan