an inverter is
NEVER a good solution if it can be avoided. inverters are bulky, inefficient, and often times cause audio/video "noise"/interference issues that can be difficult to correct. you would be much better off just getting a good regulated DC-DC PSU vs. using an inverter with an AC-DC PSU.
the PSU
may work ok as-is... hook it up directly to your battery and use a multimeter to test the rails (voltage) with the engine on and off, both no-load and under load. if the output voltage for each rail stays within +/- 5% then it should be fine. if it's way out of whack then you'll certainly need a regulated input
if you can find a "cheap" regulator that is capable of supplying 9A, then that should work fine. I suppose it depends on what you call cheap...
