Being an LCD, I highly doubt they're ferrying around analog signals. If you're lucky, the video decoding circuitry and the monitor driving circuitry are separate... if you're lucky. And even then, you'll have to reverse engineer the digital bus connecting the two, and make yourself an ADC to convert your arcade board's signals into the bitstream the monitor is expecting. If they did video decoding and monitor driving in a single custom chip, which is far more likely... game over.
Here's a far better idea. A lot of these portable DVD players have a composite video input for using the display with game consoles. Even if your unit doesn't, it may have a generic board with solder points for the header already on it. So connect a RGBHV-to-composite converter to the header, and put the player in "game" or "aux" mode. Even if your player doesn't explicitly support the feature, this may be as simple as shorting some unused solder pads where a button would go.
A harder but more guaranteed method is to do away with ALL of the player's logic circuitry and drive the LCD panel directly with some custom electronics. It's a LOT of work, but virtually guaranteed to work as long as you don't do something stupid like overvolt the TFTs. (Hint: snoop on the player's driving circuitry with a scope. This stuff is really easy to figure out if you have the right tools.)