  While the shape of the Ly$\alpha$ 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
Ly$\alpha$ profile shape and the escape of ionizing radiation within z~3
star-forming galaxies. We measure the velocity separation between double-peaked
Ly$\alpha$ emission structure (v$_{\rm sep}$), between red-side Ly$\alpha$
emission peaks and systemic (v$_{\rm Ly\alpha,red}$), and between red-side
emission peaks and low-ionization interstellar absorption lines (v$_{\rm
Ly\alpha-LIS}$). We find that the IGM-corrected ratio of ionizing to
non-ionizing flux density is significantly higher in KLCS objects with lower
v$_{\rm Ly\alpha,red}$. We find no significant trend between measures of
ionizing-photon escape and v$_{\rm Ly\alpha-LIS}$. We compare our results to
measurements of z~0.3 "Green Peas" from the literature and find that KLCS
objects have larger v$_{\rm sep}$ at fixed v$_{\rm Ly\alpha,red}$, larger
$f_{esc}$ at fixed v$_{\rm Ly\alpha,red}$, and higher v$_{\rm Ly\alpha,red}$
overall than z~0.3 analogs. We conclude that the Ly$\alpha$ profile shapes of
our high-redshift sources are fundamentally different, and that measurements of
profile shape such as v$_{\rm Ly\alpha,red}$ map on to $f_{esc}$ in different
ways. We caution against building reionization-era $f_{esc}$ diagnostics based
purely on Ly$\alpha$ profiles of low-redshift dwarf galaxies. Tracing v$_{\rm
sep}$, v$_{\rm Ly\alpha,red}$, 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.
