A conceptual answer to "exactly how and where Todd classes will appear:" You don't have to use Todd classes if you stick to Hochschild Cohomology. This might not be the best reference, but I learned this from 1.2.1 in https://arxiv.org/pdf/1804.00879.pdf You can stick to just the chern character and factor through HH(Perf(X)); the todd class corrects the fact that the HKR isomorphisms $HH(Perf(X)) \simeq H^*(X, \bigoplus \Omega_X^p)$ are NOT natural in $X$ until correction by Todd.
My apologies -- I'm sure there's a more down-to-earth reference of which I'm not aware.