So aaaaaaaaanyways.....
I'm gonna go ahead and let you be the judge the weight carrying ability of the ice. I'll also assume that you're OK with having your bike fall over.
With that aside, I've cycled through all sorts of snow and ice so I can probably shed some light on the question. When it comes to riding as you describe, it's something I'd be willing to try.
The big thing that will make or break this endeavor is the surface condition of the lake. Starting with glare ice is doable, there is more traction than people realize on glare ice, especially if you go at walking pace. You won't just sit there and spin the tire but you won't be able to balance the bike entirely on it's tires either. On glare ice your "outriggers" (feet) will be essential and they'll get a good workout. You might be tempted even to stand up and slide along as you ride but that will cause you to lose any sensation of the bike's tipping point, an before you know it the bike will be leaned over too far and slip out from under you. So keep your feet down but keep some weight on your butt too. And you can take your tire pressures way, way down, but not so much that they pop the bead.
Next would be a slightly dimpled, mottled, uneven ice surface. This can provide better grip than glare ice, and as long as the unevenness isn't big enough to catch a tire, you should be just fine at a slow pace. But any larger bumps or grooves in the ice will make it impossible to ride. The slightest rise will keep you from moving forward, and any rut will totally take control of your steering. I would suggest you do not try to ride on this type of surface.
Snow on the ice can really improve things a lot. Any snow will improve traction. Wet, heavy snow on any kind of ice surface could be OK for balancing but it's almost impossible to get any acceleration on because it's the most slippery and the least yielding type of snow. Light fluffy snow can be great, a lot of fun to plow right through it and if there's a thick enough layer then it won't matter what the ice surface us like underneath because the snow will compact itself and lift you above the ice. If there isn't a thick layer then you should still be OK but then you have to worry about what the ice is like underneath, since it doesn't take much snow to make the ice look smooth, but there won't be enough to take your weight off the ice.
Another type of snow to look out for is old snow. It will probably be well travelled, and feature tracks and windswept ridges in it, plus heating/cooling of the top layer of snow from day to night will harden it, up to the point where it's impassable.
So I pretty much disagree with everyone here as usual, except for the part about getting it on video. Oh, and tie the camera to a tree with some string in case you go under. Then we can still recover the video and get a chuckle
BTW: Here's an example of having fun on the ice with a bike. I knew the ice surface would be super smooth so I didn't put my studded tires on (until a week later) just so I could have fun trying to stay upright, or sliding along the ice;
http://www.youtube.com/watch?v=z77ci47hD9Y