  Accurate galaxy cluster mass measurements from the gravitational lensing of
the cosmic microwave background temperature maps depend on mitigating potential
biases from the cluster's own thermal Sunyaev-Zel'dovich (SZ) effect signal.
Quadratic lensing estimators use a pair of maps to extract the lensing signal:
a large scale gradient map and a small scale lensing map. The SZ bias can be
eliminated by using an SZ-free map in the pair, with the gradient map being
favored for signal-to-noise reasons. However, while this approach eliminates
the bias, the SZ power in small scale lensing map adds extra variance that can
become significant for high mass clusters and low noise surveys. In this work,
we propose projecting out an SZ template to reduce the SZ variance. Any
residual SZ signal after template fitting is uncorrelated with the SZ-free
gradient map, and thus does not bias the mass measurements. For massive
clusters above $4\times 10^{14}$ $M_{\odot}$ observed by the upcoming CMB-S4 or
Simons Observatory experiments, we find that the template fitting approach
would increase the cluster lensing signal-to-noise by a factor of 1.4.
