Physical-layer security in free-space optical communications channels can be compromised when an eavesdropper performs optical beam-splitting attacks over an atmospheric channel. Previous simulations have shown that Laguerre-Gaussian orbital angular momentum-carrying beams can provide higher secrecy capacities compared to that of ordinary Gaussian beams. In this paper, we determine if Bessel-Gaussian beams can provide further improvement over their corresponding Laguerre-Gaussian counterparts. Using computer simulations and experiments with spatial light modulators, an increase in secrecy capacity of 10 to 30 bits/sec/Hz in the weak to medium turbulence regimes is demonstrated. This verifies that Bessel-Gaussian beams have more resiliency to atmospheric turbulence effects than Laguerre-Gaussian beams. Furthermore, research on optimizing the quality of these beams can help to realize a practical system for more secure communications.