@rob, not sure I’d say I was an expert ! On your MBP do you have have the airplay logo on top line of home desktop screen (I assume you are running mavericks) ? If so you can connect to ATV and use the tv as a second screen complete with dock (all the program icons on the bottom). I suspect you can as you have sound (note some older macs don’t support this like my mac min from 2009). What you want to do is get this second screen up on the tv. Then scroll mouse to tv screen and start the DVD app from there, max the screen and you are done.
If this doesn’t work you need to rip the DVD (via handbrake or RipIt) and import into iTunes and AirPlay it from there.
Good luck
Edit FYI the programme RipIt will get past pretty much any copy protection, I use handbrake to compress the result and put it imto a mac format like mkv, I find handbrake doesn’t always get past all copy protection.