How does the amount of liquid in a cylindrical can affect its motion when rolling down an inclined plane?

If you are dealing with water, then it will behave nearly like an inviscid fluid, meaning that, aside from a very thin boundary layer near the wall, the fluid will not be rotating as the can rolls down the incline. Basically treating the water as inviscid is equivalent to allowing the water to slip at the wall. This would prevent the can rotation from propagating into the fluid. So the water would move down the incline as a non-rotating body while the can would rotate around it (and would have rotational inertia). The problem could be modeled that way.

A very highly viscous fluid, on the other hand, would behave as a rigid body stuck to the wall whose center of mass is offset from the axis. It would be possible to model this situation also.

Another situation that might not be too hard to analyze would be a moderately viscous fluid that fills the can.

However, other situations would be pretty difficult to get a handle on. For example, the case of a moderately viscous fluid that does not fill the can.

In my judgment the range of possibilities of % fill and fluid viscosity is too broad to be tractable right now. I think you need to narrow the range of possibilities that you are willing to consider.