How you get there.... Well, FAA thinks differently  

To reduce the burden of the commute and save its crewmembers
the daily parking fees at MSP, Red Wing Aeroplane offers its crewmembers free
transportation from the company's offices in Bay City to MSP. The crewmembers may opt
to commute on their own from their residence [which would include a perk free quarters to save the pilot money] in Bay City, Wisconsin, or may opt to be
driven to MSP by Red Wing Aeroplane. You question whether the hour commute to MSP
constitutes transportation that is local in nature such that it may be considered part of the
crewmember' s rest period.

Based on the facts as outlined above and the following analysis, the FAA has determined
that the transportation provided by Red Wing Aeroplane could be considered transportation
that is local in nature, and thus, part of the crewmember's rest period as long as there is no
present requirement for duty at the company facility that would effectively end the
crewmember's commute for purposes of applying the local in nature definition as described
below.

https://www.faa.gov/sites/faa.gov/files/faa_migrate/interps/2011/Converse-RedWingAeroplane-2_2011_Legal_Interpretation.pdf

Your duty period is at the base not at the free pilot quarters provided as a perk.   How you get to/from the base is up to you.  The quarters are a perk.   How you get to and from your perk is on you!

If you feel the company is in error and violating FAR 135.263, you can always whine to the FAA on their hotline or safety reporting system to prove them in the wrong.  But, likely, that will result in the perk going away.  

Even if it was from a hotel/motel, the commute would be on you!