Can BCP Out cause resource contention
I would guess these symptoms (PAGEIOLATCH_SH waits and longer execution times) are simply due to increased physical IO. SQL Server has to scan the entire source table during the BCP which may evict pages from buffer cache that the stored proc later references. When run by itself, the pages needed by the proc probably remain in memory.

Your assumption about BCP needing to read the entire table into memory before writing the result is incorrect. BCP streams results from the source query to the output file.