"The LOC needs no course info, it is a pure fly L/R command and does not need heading integration as does the VOR. "
According to the Boeing Maintenance Manual, the A/P uses the selected CRS to refine its intercept calculations. For those with access to the manuals, use "delta runway track" as a search phrase.
The capture arc varies depending on the initial intercept angle. It may be a two arc capture if the angle is too great or too small.
Rgds.
NSEU