Hmmm

If it wants 7-12V then you can run a 9V battery a lot flatter than a 7.2V battery before it drops below 7V. The poor things do tend to fade before they reach their rated Ah.

I think it's looking for a 9 Volter.

Is it really drawing 400mA?

0.4A at 9V is 3.6W. Sees like a lot, does it run hot?