I've never heard of cockapoos being a dominant breed. Poodles and cockers are not knows for it. Poodles are very smart and a living family dog. Cockers are a hyper but kid loving easy to get along with breed. Mixing them doesn't change what they are.

But in the breed, wether chi, dobe, cockapoo or great dane, there can be lazy or hyper, aggressive or push overs, it just seems like you have more of a pushy dog is all. No worries, 'Lo is a high energy, bossy brat with a barking problem lol.
Quoting the best from
http://dogs.about.com/od/funphotos/qt/tugowar.htm
"Many dogs love to play tug of war - it is a healthy display of their predatory nature. There has been some debate over tug of war and dogs. Some feel the game causes aggressive behavior and dominance. However, the tug of war is actually mental and physical exercise for your dog. It is also a great way to reinforce the human-canine bond. The most important thing to remember is that there are rules that should be followed to keep the game from going overboard.
1.) Before you begin playing tug of war with your dog, you should teach a release command, like drop it. This will help you stop the game if necessary.
4.) While playing tug of war, your dog might get excited and begin growling. This is normal, as the game itself is predatory behavior. However, it is important to keep your dog from becoming overly excited. A low mild growl with tail still wagging is okay, but anything more warrants a break. To me (not quoting here!) this means a direct stare, a very tense body and still tail, or grabbing the toy to hard and forcefully. Return to quote.
5.) To take a break, stop tugging and use the release command. Take 30 seconds or so to go through basic commands like sit and down. Once your dog seems more relaxed, the game may resume.
6.) If your dog's teeth come into contact with you at any point, play should stop immediately. Let out a yelp, say "drop it," then take the toy and walk away for 30 seconds. If your dog seems relatively calm, you can go through the break with commands, then begin the game again. If your dog makes the same mistake two or three times, tug of war should be ended for the day. This is just to remind your dog to be extra careful with her teeth. It is likely that teeth might graze you from time to time due to the nature of the game, but once your dog understands the rules, she will be much more careful.
6.) While playing tug of war, it is okay to let your dog win! This builds her confidence and rewards her. However, if she misbehaves, you should be the one who ends up with the toy." end of quote.
I do agree with 6, when 'Lo and I play tugofwar, towards the end I'll rev her up, get her growling and pulling and tossing her head, give it one last pop and let it go. She'll jerk back with her prize, toss her head, parade around grinning, then sit down to a good chew. But if I ask for something, she never hesitates, what I want is mine, and she knows it believe me!
You can let them win, and still stay in control. Tug of war is a game, and just that, a game! Dogs play rough together, and rules are ignored because it's a game. The normally bossy dogs will be rolled over, chewed on, sat on, everything! Out of the game he/she will not be messed with, and the other dogs will listen again.

But you can play with him, and let him win, you'll be a better leader (a fun one!) not a weakling.