Is there an algorithm for minimizing an NFA with respect to bisimilarity rather than language equivalence?
Yes, see for example here:

Johanna Högberg, Andreas Maletti, Jonathan May,
Backward and forward bisimulation minimization of tree automata,
Theoretical Computer Science 410(37), 2009, pp. 3539-3552,
https://doi.org/10.1016/j.tcs.2009.03.022