shared/install: consistently use 'lp' as the name for the LookupPaths instance

shared/install: consistently use 'lp' as the name for the LookupPaths instance

Most of the codebase does this. Here we were using 'p' or 'paths'
instead. Those names are very generic and not good for a "global-like"
object like the LookupPaths instance. And we also have 'path' variable,
and it's confusing to have 'path' and 'paths' in the same function that
are unrelated.

Also pass down LookupPaths* lower in the call stack, in preparation for
future changes.