Yes, you'll need a battery. The battery also acts as a filter in the battery charging circuit. In fact, many older charger/converter's will not work without a battery installed. You can test the health of your converter by measuring voltage between the battery terminals, once installed and connected to shore power. The voltage should be greater than 13.2 V.
If you don't have a handheld electrical meter, get one and learn how to use some of it's functions. They are inexpensive and a "must have" for diagnosing problems and maintaining a travel trailer.
You could probably get by with a cheap battery if you always have shore power available. Some of us camp off-grid so have banks of batteries and solar to keep everything running.
One dirty little secret is that most trailers have parasitic "phantom" loads that will suck a battery dry in a week or so if the battery is left connected. IMO, every trailer should have a battery disconnect switch for storage. This is one item that the Mfgs. choose not to install because of greed. If you park the trailer with your new battery and you don't connect to 120V AC and don't know if the converter is working, disconnect a wire from the battery terminal. A disconnect switch is a better solution.
Good luck with your new rig and buy that meter!