I would think that
FESTE would be your best bet - they operate a database of all RSC productions, hosted on the Performing Arts Data Service website - click on this link:
http://appserver.pads.arts.gla.ac.uk/cgi-bin/WebOb
jects/FESTE
If you're looking for the casts for, say, all plays performed at 'The Other Place' in Stratford, just type 'THE OTHER PLACE' in the 'THEATRE NAME' section (by default it should say 'STARTS WITH' on the dropdown menu, then click 'MATCH'. It will then list
all the productions performed there since records have been kept (n.b. not just Shakespeare plays). Find the production you want and then click 'CAST AND REVIEWS' and Robert's your father's brother. The Swan is listed simply as SWAN btw and The Royal Shakespeare Theatre simply as ROYAL.
Hope that helps!