Maximum fractional packing of spanning trees.
The problem gets very easy if you look at the feasible space, which is simply the spanning tree polytope, intersected with a linear number of half-spaces, one for each packing constraint. 

The spanning tree polytope has a separation oracle (check this for example), therefore since you only have an additional linear number of constraints, it carries over to yield a separation oracle for your problem. Using this oracle, you can optimize any linear functional of the polytope in polynomial time.

If I remember well, using the ellipsoid algorithm, you can get a running time of $\tilde{O}(m^2 \cdot T)$, where $T$ is the running time of the separation oracle (which in this case you can get from a min-cut algorithm). Probably not as good as other existing algorithms, but it gives you an idea of how you can quickly check that the problem is polynomial time solvable.