Steve-
Gerald and Leah are absolutely right. That high wattage heater will quickly run down your coach (house) battery bank off the inverter, and is one appliance that shouldn't be used that way, and during your test you probably just forgot it was on.
Usually your house batteries consist of four 6 volt deep-cycle ones, designed to hold up against regular discharges and recharges, and each offers more amp-hours than a 12 volt deep-cycle coach battery. Starting (chassis) batteries aren't built to cycle like that.
The four 6 volt batteries are in two sets; each set of two are wired together serially (serial: one added to the other) as a pair to provide 12 volts total. Then the two sets are wired in parallel (two 12 volt systems side by side), to provide plenty of 12 volt amp-hours of total service.
Look at it this way, volts X amps = watts, so amps = watts/volts. That is true for both DC (12v.) and AC (110v.). The only thing that stays the same in either voltage source for an appliance load is the watts. So if your heater or a hair drier is say, 1500 watts, it will draw maybe 14 amps plugged into 110v. current (1500w /110v = 14a). But if that 110v. current is being inverted from a 12v. bank of batteries, your socking them with 125 amps (1500w/12v = 125a). Ouch! The voltage left to push that kind of current will quickly drop.
If you know the wattage of each of your devices and appliances, you can calculate how much draw each will put on your battery bank when on the inverter. For example, the digital clock on the microwave won't take much juice, but nukeing food with the 1000-1200w appliance will.
-Joel