Computing a feasible flow doesn't seem terribly complicated. For each edge on the graph you can find a path that contains it. Then start with 0 flow for every edge, loop over the edges and add the minimum flow for the each edge to the path that includes this edge. I am not sure if this can be done in "single traversal", because it's late and because it depends on exactly you mean by that. But I think it will be faster than solving another max-flow problem (although I didn't fully follow what you posted).
See, I was thinking that finding a feasible flow like this has really bad running time but I think it doesn't at least not relative to max flow. You could just do a complete depth-first traversal and then when you either reach t or reach an already-discovered vertex augment the current path you are on with the max lowerbound of the set of edges that were newly discovered on the current path, going all the way to t each time you do this. This would have running time of something like O(E * Avg. length of paths from s to t).
The algorithm involving two max flows is what you need to do when edges have both lower bounds and upper bounds, as someone is discussing in this power point file.
There also turns out to be a small amount of literature on this problem itself -- that is, minimum flow -- not just as a use of max flow or as formulated as minimum cost: see here and here, for example, but all of this material deals with the case in which there are lower and upper bounds. It actually surprises me that there is nothing in the literature about the no upper bound case; it seems like a very practical problem.