To echo David's experience, I think I've gone through close to 20 airport security screenings both domestically and internationally with an OpenROV as carry on and it usually goes just fine. Very often, they ask what the thing is and open the case and do a swab of the ROV. Whenever I describe that it's an underwater robot, the people doing the check always seem to perk up and get excited about the project. I swear, we could probably do pretty good business just selling ROVs to people who we meet in TSA lines!
As for batteries, there are some funny rules about not being able to take individual batteries on planes, but if the batteries are inside of something (like your ROV) then it's okay. The whole thing seems a bit fuzzy and I'm still not sure of an exact policy, but in general I've never had trouble with bringing ROVs with batteries inside on as carry on, nor have I had trouble bringing spare batteries along with the ROV.
Best of luck with your travels, and send some photos of the trip!!