I don't have a 32bhds, so I don't know if the TV works on DC and AC or just AC. My guess, however, is that the TV only works on AC (ie, when you're plugged into shorepower/electrical pedestal. During your walk-thru, the TV worked b/c the trailer most likely was plugged into shorepower/electrical pedestal. But since you state that there is a 12V DC connection under the TV, if you're not plugged into shorepower/electrical pedestal you can still operate the TV, provided however, that you obtain an inverter with sufficient wattage to run the TV (an inverter is a device that changes/inverts DC current into AC current). You can purchase such an inverter at a big box store or online, where one end plugs into the 12V DC connection (eg, cigarette lighter type) and then you would plug your TV (or any other AC required devices) into the inverter. If you only intend to run the TV and perhaps the stereo, figure out the number of watts needed for the TV and stereo. For example, let's assume your TV requires 100 watts and your stereo requires 80 watts. That's 180 watts; for simplicity, let's round-up to 200 watts. If inverters were 100% efficient, you would simply need an inverter capable of providing 200 watts. Unfortunately, inverters are not 100% efficient, so let's increase our 200watts by 20% (which pertains to the inefficiency of an inverter) and that gives us 240 watts (200 x 20% added to 200). So, at a minimum, you should look for a 250 watt inverter; I'd recommend a 300 watt inverter for this example. That's it; get a 300 watt inverter, plug into your 12V DC connection and you'll be able to power-up the TV or stereo when not plugged into shorepower/electrical pedestal (assuming that your battery bank is connected).
If you decide to run the TV and/or stereo off your battery bank via the inverter, keep in mind the amp draw. For instance, using the above numbers, if your TV requires 100 watts, that equates to 8.3 amps (Amps x Volts = Watts, where Volts=12 and Watts=100). If you run the TV for 3 hours, you'll consume about 25 amp/hrs (8.3amps x 3hrs) (note: it actually is a bit more than 25amps because the inverter needs some DC amps to work). You will need to recharge your battery the next day assuming that you have the basic 105 amp/hr battery, where you don't want to consume more than 50% of the amp/hr rating (which would be about 52.5 amp/hrs). In other words, just running the TV for 3hrs doesn't leave you with much DC power for other DC devices.