Out at the lake a few weekends ago, my best friend and I were goaded into a "king of the dock" competition by our kids. He is very big and strong, and I am very average. He doesn't know aikido.
During the contest, I was very wary to use any of my aikido skills, as I didn't want to dump him onto the corner of the dock (or get dumped myself). Indeed, that happened later with my son and his son and my son was bruised pretty bad -- a full sized adult would have been messed up.
So we pushed each other respectfully and his strength and size won out in the end. I think my kids were disappointed because they wanted to see the aikido instructor do some magic! I don't know what would have happened if I had gone into full aikido mode to win the contest (that even sounds weird), but I think there would have been an injury. I would have felt awful hurting my best friend and ruining our cabin getaway.
Situations on the mat are very much like this. We train cooperatively because the techniques can be very, very dangerous and we don't want to hurt each other. When someone wants to "test" me or compete with my technique, I do the same thing as when I was on the dock that day -- I let it go (whether they know it or not). I may look bad (or not), but it's a small price to pay to save them from getting hurt.
As we gain deeper insight into aikido, these situations become less frequent and less intense, to the point where we are at the level of your sensei who can still execute a technique safely against a large uke who is resisting. I have seen (and felt) this from senior teachers time and time again, so I know it is possible with continued training. I can also do it quite often against people that are only a little bit bigger than me or not resisting too much!
In the meantime, struggling to force a technique to work or doing something stupid like hitting the guy is just going to create conflict in the dojo. That kind of training is not only dangerous, but also very unpleasant.
I believe O Sensei said that aikido is always life and death, not to be fooled around with. When my students show interest in "playing" or competing with each other, I try to kindly remind them of this fact.
Having said that, dedicated study of kaeshi-waza between senior students is a completely different matter. I think most people would easily grasp the difference in tone and intention between the two situations. At 5th kyu I think your sensei should tell the big guy that he needs to develop sensitive ukemi. Remember, no matter how big you are there is always someone bigger.