I have not seen any Bayesian factor counterpart to the McNemar test. However, there are some Bayesian counterparts which can be viewed as the McNemar test.
Keep in mind that there are several variations of the McNemar test, each of which has their own merits/pitfalls.
For the exact McNemar test, which is arguably the most widely-used test in this family, there are several solutions. One simple approach is to use the beta-binomial conjugate to simply calculate the posterior probability)
One drawback of the beta-binomial conjugate is that it only considers two states from events A and B: either A > B and A < B. So, this test throws away the samples that A and B have the same value (both success or both fail.) To take into account the equality of A and B, you should use the Dirichlet-trinomial test which can be find here.
The third approach is to use the Bayesian Sign test. If you set p=0.5 in the McNemar exact test, then it is equivalent to the Sign test. The Bayesian Sign test's implementation can be found here.
I hope this helps. Unfortunately, I could not post more than two link. You can google various frequentist McNemar test (asymptotic, asymptotic with continuity correction, exact, mid-p) and you can find suitable resources for the beta-binomial conjugate.