Online set cover variant? Routing of requests
In general, there are two approaches you can take here:

Empirical evaluation. Pick a realistic workload (a data set with a bunch of scenarios, where each scenario is a sequence of requests).  Use this as a benchmark.  Test each candidate heuristic on that benchmark, and see which one performs the best on this benchmark.

Theoretical analysis. Alternatively, you can try to prove a theorem about the competitive ratio of these two heuristics.  That may be very difficult.

I suspect it's going to be a lot more feasible -- and potentially more useful -- to take the former approach.  It does require you to have a dataset of representative requests.



I can suggest another plausible heuristic.  Given a graph, you can find use bottleneck shortest paths to find the best path from $s$ to $t$, where "best" is measured by the number of slots in the least empty node that's visited along that path.  The larger that number, the better (it means there exists a path with plenty of spare capacity).  So, one possible heuristic is to choose whichever path reduces this number the least.

There are probably many other heuristics you could come up with.  I suggest brainstorming a few, then evaluating them empirically using the approach listed above.