If you can get your hands on a August 2014 Trailer Life Magazine, there is an excellent article on Converters starting on page 46. It mentions 4 manufacturer "upgrades" to 4 stage smart converter/chargers.
If you need a LOT of DC amps, the Xantrex True Charge 2 can be "stacked" to work together to provide up to 120 amps of 4 stage charging power and concurrent usage.
Now, for your problem:
Have you checked the INPUT wires to your converter for 120 volts AC?
I would verify that you do not actually have a bad circuit breaker or feed wire to the unit.
It is no surprise that a near dead battery will recover enough between uses to power a very low draw item for a few seconds and still fail to generate enough amps/volts to even budge a higher draw one.
When low in number (near dead), free electrons deep in the lead plates require time to move from the center of the plates to the surface of the surface to be available for use. Happy where they are, they will need "demand" (closed circuit) to even start moving. When packed to the gills with free electrons (fully charged), the electrons are readily available for use from the surface and in large quantities without calling up the reserves inside the thick plates.
This will only happen for a short time (in a deep cycle battery) before reserves from inside the plates are needed. The ability to supply large amounts of power drops off quickly when that happens since deep electrons can't all get to the surface at once (known as the Peukert Effect).
Starting batteries have huge amounts of surface area, but are very thin. While they can provide large quantities of starting amps, they have little available "in reserve." They can start the coldest engine, yet kill the battery just playing the radio for a few hours while you wax the car (old days before automatic shut offs to protect the battery).
(Layman's explanation - scientists take note)
Lou and Laura with Bella - German Short Hair Pointer
2008 GMC Sierra 2500HD Crewcab SB Allison Duramax
2010 Flagstaff 8526RLWS - Superglide 3300
HAM CALLSIGN - KC3FFW