I can't imagine anyone would remove the factory installed converter either, unless it failed. But RichR told you EXACTLY how to test if you are receiving a charge on the battery when plugged into shore power (or, when plugged into your tow vehicle with its ignition on, for that matter). You need to use a multimeter and check the voltage at the battery terminals. If you don't have one, borrow one from a friend or let the friend test this for you.
If the trailer is unplugged from a tow vehicle, your battery trickle charger, and shore power and everything in it is turned off, the battery voltage will probably be no higher than 12.7 V if everything has been that way for at least 24 hours. This is known as "resting voltage" and is what you see on a battery that is in good shape. If the voltage is significantly lower, say 12.3 V, the battery is probably weak and could need to be replaced. If it is higher than 12.7 V, it may not have been unplugged from a charging source long enough.
Now, plug your trailer into shore power so that the converter is getting power. Test the voltage at the battery terminals again. If it's higher, in the 13.5+ V range, your battery is being charged. You should see the same thing when plugged into your tow vehicle WITH THE ENGINE RUNNING! If not, your tow vehicle is not charging your trailer. And, of course, if you put your trickle charger on your battery, it should raise the voltage the same way.
A multimeter is a "must-have" troubleshooting device for electrical problems! You can find inexpensive ones (about $30) that are reliable at Home Depot or Ace Hardware. Stay away from the "super cheap" (less than $10) ones at Harbor Freight.
- Jack