MACSE (Hungarian Society for Family History Research) www.macse.hu is an excellent resource for post 1895 Budapest vital records. Its database is compiled by volunteers, so it is incomplete, but it is definitely worth a look before you go to the trouble of contacting the City Archives of Budapest. The database is compiled from the civil records that have been digitized on familysearch, but the MACSE transcriptions are not available in the familysearch website.
MACSE is fee for service ($30/year) and it can be a bit tricky to sign up (I can give you the email of the person who helped me).
If you want to contact me privately, I'd be happy to do a short lookup and let you know what kinds of records, if any, are available for your relative.