While the shape of the Lyα profile is viewed as one of the best tracers of ionizing-photon escape fraction (f esc) within low-redshift (z ∼ 0.3) surveys of the Lyman continuum, this connection remains untested at high redshift. Here, we combine deep, rest-UV Keck/LRIS spectra of 80 objects from the Keck Lyman Continuum Spectroscopic Survey with rest-optical Keck/MOSFIRE spectroscopy in order to examine potential correlations between Lyα profile shape and the escape of ionizing radiation within z ∼ 3 star-forming galaxies. We measure the velocity separation between double-peaked Lyα emission structure (v sep), between red-side Lyα emission peaks and systemic (v Lyα,red), and between red-side emission peaks and low-ionization interstellar absorption lines (v Lyα−LIS). We find that the IGM-corrected ratio of ionizing to nonionizing flux density is significantly higher in KLCS objects with lower v Lyα,red. We find no significant trend between measures of ionizing-photon escape and v Lyα−LIS. We compare our results to measurements of z ∼ 0.3 “Green Peas” from the literature and find that KLCS objects have larger v sep at fixed v Lyα,red, larger f esc at fixed v Lyα,red, and higher v Lyα,red overall than z ∼ 0.3 analogs. We conclude that the Lyα profile shapes of our high-redshift sources are fundamentally different, and that measurements of profile shape such as v Lyα,red map on to f esc in different ways. We caution against building reionization-era f esc diagnostics based purely on Lyα profiles of low-redshift dwarf galaxies. Tracing v sep, v Lyα,red, and f esc in a larger sample of z ∼ 3 galaxies will reveal how these variables may be connected for galaxies at the epoch of reionization.