Back in college, I did some downhilll racing at a ski resort with chairs like the ones you're talking about.
One guy carried his bike like this:
It had the advantage that when he dismounted, he just set the rear wheel down while still sitting in the chair, stood up, and walked off the lift with the bike next to him. The disadvantage was that it was an awkward ride to the top. If I remember correctly, it was also a little awkward to get onto the lift. He just sort of muscled it all into place.
Unfortunately, I don't have a picture but most of us did it differently. We would straddle our bikes just like we were about to mount them while waiting for the lift. When the ski lift came up behind us, we would pick our bikes up and set the handlebars across our laps with the frame between our legs. As we got lifted up, the bikes would just dangle below the lift chair. Then, to dismount, we would just set the rear wheel down, stand up straddling the bike like when we got on the lift, and take a couple of awkward steps forward, just like you might do if you stop a little short at a stop light or something.
This way had the advantage of being super easy to get yourself and the bike onto the lift, as well as being easy to hang onto your bike on the way up. You could also fit two people on the chair at the same time. On the flip side, it was awkward to get off the lift, but dismounting never felt dangerous at all.
Honestly, the ride to the top was super scary (for me, at least). As you can see from the picture, there's no safety bar on those lifts. You could feel the weight of the bike causing the chair to lean forward a bit. I preferred to lift my bike a little higher than everyone else and rest the seat on the edge of the chair between my legs. This allowed me to pull the handlebars back closer to my face, balancing the whole chair out a bit. It was a little awkward to hold the bike like that, but the lift chair felt more stable. It clearly wasn't necessary since no one else did it and no one fell off the lift, but it made me feel better.