So it's actually the Destination setting for exporting PDFs that will control how the PDF is ultimately printed?
No, it's the actual press conditions that determine how it prints, and without any knowledge of those, it's a crap shoot. If your Destination Profile (U.S. Web Coated (SWOP) v2?) is embedded in your PDF as the Output Intent (which will happen if you're sending PDF-X and that profile is assigned to your document) the printer may use that as a source from which they will convert to their actual press profile. But more likely, they'll just print the values in you give them, so if you're working with a profile that specifies 15% dot gain for the midtones, but they're actually running at 25% dot gain, your midtones will print 10% darker than expected.