Gravitational lensing of the cosmic microwave background (CMB) generates $B$-mode polarization that acts as a source of contamination to searches for $B$ modes generated by primordial gravitational waves (PGWs). The strongest constraint on PGW $B$ modes is already significantly limited by lensing $B$ modes, as shown in the most recent BICEP result. In this work, we present CMB lensing $B$-mode templates constructed using SPT-3G and  data, which characterize the lensing $B$ modes and can be used to improve PGW $B$-mode searches. We use SPT-3G data from the 2019 and 2020 observing seasons for the $E$ modes and the CMB-reconstructed lensing potential, and a cosmic infrared background (CIB) map from  as an external lensing tracer. To test for extragalactic foreground biases in the lensing template, we consider CMB lensing reconstruction variants with different levels of foreground immunity: the standard and profile-hardened global minimum variance (GMV) quadratic estimators, and a polarization-only quadratic estimator. We validate the template construction using Gaussian simulations and  simulations with realistic non-Gaussian foregrounds. From simulations, we find that foreground-induced biases are strongly suppressed for the template constructed with the profile-hardened GMV + CIB tracer, with residual bias below 10% of the statistical uncertainty on the template power spectrum. Data difference tests on this template similarly show no evidence for significant foreground contamination. Using simulations, we predict that this foreground-immune lensing template achieves a residual delensed $BB$ power of $A_{\rm lens}^{\rm res} \simeq 0.48$ averaged over $20\le \ell \le 200$, the highest delensing efficiency of a lensing template to date. These results demonstrate and validate a method to construct foreground-robust lensing templates which will be used in upcoming delensed PGW $B$-mode analyses of BICEP data.