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While the shape of the Lyα profile is viewed as one of the best tracers of ionizing-photon escape fraction (fesc) 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 (vsep), between red-side Lyα emission peaks and systemic (vLyα,red), and between red-side emission peaks and low-ionization interstellar absorption lines (vLyα−LIS). We find that the IGM-corrected ratio of ionizing to non-ionizing flux density is significantly higher in KLCS objects with lower vLyα,red. We find no significant trend between measures of ionizing-photon escape and vLyα−LIS. We compare our results to measurements of z~0.3 "Green Peas" from the literature and find that KLCS objects have larger vsep at fixed vLyα,red, larger fesc at fixed vLyα,red, and higher vLyα,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 vLyα,red map on to fesc in different ways. We caution against building reionization-era fesc diagnostics based purely on Lyα profiles of low-redshift dwarf galaxies. Tracing vsep, vLyα,red, and fesc in a larger sample of z~3 galaxies will reveal how these variables may be connected for galaxies at the epoch of reionization.