A Karl has said, the problem is almost surely that the initial power draw from turning on the tv is large enough to overload the inverter. It's possible a larger inverter will help, but I am more inclined to think it is a 12V input problem. When the tv/dvd powers on, the initial surge causes the inverter to suck a lot of amps from the battery and the input voltage drops. That in turn causes the output (AC) voltage drop and the tv desn't complete its power on cycle. The inverter might even shut itself off if the input voltage drops low enough (typically around 11 VDC).
There are a couple of likely causes for this problem. First of all, standard cigaraette lighter outlets are often underpowered, with wiring that is too small a diameter and too long a distance to the batteries. This can be correced by running a large gauge wire (12 gauge will handle the max that a lighter plug can deliver) as direct as possible to the batery itself or the 12V distribution (fuse) panel.) The other likely cause is poor batteries or a corroded connection to the batteries. The initial battery voltage may be low or their condition may be poor enough that they cannot sustain their voltage when any load is applied. [Batteries sometimes show a good voltage at rest but "die" under any sort of load].
How old are the batteries? Have you checked the water in the cells lately? With your genset and shore power off, put some 12V lights on in the rig for 40 minutes or so, then check the DC voltage at the lighter plug or fuse panel. If less than about 12.5 volts, I'd say your batteries are in poor condition.