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 homology. 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.