An alternator puts out high frequency AC, which is rectified to DC by diodes, but there is still a ton of ripple (not "happy" for most electronics, like the inverter's input). Also, the field coil of the alternator needs something to pull power from to get the generation running - once going, you could remove the battery and have it regulate off the voltage it generates, but the regulator will not be at all tolerant of sudden engine RPM changes, or big load changes (like an air-conditioner starting up).
You need a battery of some sort if you plan to do DC power at all, however, you might be able to reduce the size of the battery bank by using ultracapacitors - it's just not how alternator regulator manufacturers envisioned the system when designing the regulator, so your functionality may vary (and require a good amount of tinkering/experimenting).