In almost all cases I know of where people have proved derived equivalences between blocks of finite groups, the proof hasn't really gone that way (i.e., finding a virtual bimodule and refining it to a splendid tilting complex). In fact, usually the virtual bimodule doesn't appear explicitly at all, although in most cases it would probably be possible to calculate it if you wanted.
There are two exceptions I can think of, where there was an "obvious" choice of a virtual bimodule, and later it was proved that there was a corresponding derived equivalence.
The first example arises for blocks of symmetric groups. About 30 years ago, I noticed a virtual bimodule that gave a virtual Morita equivalence between certain pairs of blocks of symmetric groups, such that if these could be lifted to derived equivalences then all blocks of symmetric groups with the same defect group would be derived equivalent. I found a natural way to turn them into a complex, but couldn't prove much. But later, Chuang and Rouquier proved that it did work in:
Chuang, Joseph; Rouquier, Raphaël, Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification., Ann. Math. (2) 167, No. 1, 245-298 (2008). ZBL1144.20001.
The second example is "Alvis-Curtis duality" for finite reductive groups in non-defining characteristic. Broué had conjectured that this was given by a derived equivalence, and there was an obvious choice of virtual bimodule (and a fairly obvious way to make it into a complex), which Marc Cabanes and I proved to work in:
Cabanes, Marc; Rickard, Jeremy, Alvis-Curtis duality as an equivalence of derived categories, Collins, Michael J. (ed.) et al., Modular representation theory of finite groups. Proceedings of a symposium, University of Virginia, Charlottesville, VA, USA, May 8-15, 1998. Berlin: de Gruyter. 157-174 (2001). ZBL1001.20002.