With the symptoms you have described I think you can rule out a blown fuse on the output side of the battery. My guess would be is the battery is not being charged properly by the converter or the battery is failing under high current demand. Since everything seems to work good when you disconnect the battery that would indicate to me that the converter is working properly. When the battery was checked did they check it under load?
If the DC lights and appliances work ok when the battery is not connected that seems to indicate the converter is doing its job but somehow that charge from the converter is not getting to the battery or the battery is bad. I see no reason to replace the converter if it will run the dc stuff without the battery.