Let me make sure I understand some details. You are solving the route against some kind of cost attribute that is either time or distance. After the route is solved, you want a way to accumulate what tolls were paid along the way. You don't care, necessarily, to avoid tolls, you just want to know how much was paid along the way.
Your toll information is stored in a table with edge IDs? This is as opposed to using junctions, turns, or point barriers for tolls.
If I have the details right, here are a couple of suggestions:
Set up a cost attribute on your network, based on tolls paid. Call it TollCost or something. When you solve your route against your regular cost attribute, say TravelTime, you can also accumulate TollCost. You should get a Total_TollCost value with the output route.
Another idea, if you really want to use Traversal Result or accumulating doesn't work or you don't have access to edit your network attributes, would be to change your routing service. Instead of just using the traditional route service, set up a GP model that solves a route, generates a traversal result and perhaps does some analysis on tolls before returning an answer.
The best answer really depends on how you model the toll costs along with performance and client/server processing expectations.
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