Hi,
From your description, you may have a bad battery, a bad converter, or both.
If your camper uses a 12-volt lead acid battery, these would be the voltages you can look for. Ideally, you need a digital voltmeter with at least 1, but 2 decimal places are better. Yes, you have to look for at least one decimal place. There is not a lot of voltage change, but you can see differences to tell what is going on.
Assuming the camper has been plugged in and the converter was on, you can start here.
Unhook the battery negative wire, or turn off the battery disconnect, or unplug shore power to remove the power converter from the battery. Test the battery terminals with the meter. What is that voltage? This is a starting place. Since the power converter was on, the battery may have a surface charge on it. More on this surface charge later.
Now, hook battery the back up, or turn on the disconnect, or plug in the shore power. Now test at the battery terminals. These voltages should be in this range.
13.25 VDC = float charge. (if your converter has float mode)
13.65 VDC = standard charge.
14.4 VDC = boost charge. (if your converter has 3 stage charging)
The readings may be +/- 0.10 approx. from the above target points. They should not be lower then 13.0/12.9 or a lot more then 14.5/14.6 VDC, if they are lower or higher, there is a problem.
After testing the above, here is a battery test. Unhook the battery from the converter by what means you have the easiest way. Take a battery voltage test at the terminal. Record the number. Then wait, over night or at least 8 hours, take another battery voltage test. Record the number. Waiting the 8 hrs lets any surface charge drain off and then the battery is at rest. Here is a few voltages for a battery at rest for a lead acid battery.
12.73 VDC = 100% state of charge
12.10 VDC = 50% state of charge
11.81 VDC = 30% state of charge
As you can see, you need the decimal place as 1 volt is a lot of change. Your battery might read above 12.73 VDC (13.0 plus volts) when it just came off the power converter, but that higher voltage will drift down over the 8 hours of non battery use and you get the battery at rest voltage which is what you are after.
See this link to the Trojan battery maintenance for a complete chart on voltage. Scroll to the "II Open Circuit Voltage Test" https://www.trojanbattery.com/resources/battery-maintenance
A power converter going bad, can fail in a few ways, it goes high voltage and boils out the battery killing the battery. It stops working all together and no longer charges the battery. It goes low voltage, this then hurts the battery draining it below its fully charged state. When the power converter goes bad, it can kill a good battery. And a bad battery, may or may not hurt a good working converter.
If you tell us the brand and model converter, the year and make of the camper, the voltage readings from above, we can help better. And confirm what battery type you have.
Hope this helps
John