is there a way to use the graph-tool algorithms to test whether a connected
component of an undirected graph is 2-edge-connected, i.e., none of its
edges is a bridge which, if removed, would disconnect the component?

Of course one can do that by filtering out edge by edge and labeling the
components of the resulting graphs, but that is probably not the most
efficient way.

The algorithm for 2-edge-connectivity is simple, and think can be
implemented in time O(E). If you open an issue for this in the website, I'll
implement it when I find the time.