Yes the converter is a charger also. An electrician co-worker told me one time that a deep-cycle battery needed to be charged harder (higher amp) to get them charged.
When charging on a charger it it best practice to pull the vent top off to let it vent better (also lets you see the water (acid) level).
If you can plug into either 120v OR 12v TV (as long as it is wired to power the trailer) you SHOULD be able to move the slide with a low battery.
Steve & Sandy
2006 GMC Duramax-Edge Evolution