I finally figured it out! The 11g R1 32 bit client succeeded where the R2 client failed. I cleared out the existing installations, installed the 32 bit 11g R1 Oracle Client followed by the 64 bit 11g R2 Oracle Client. I considered doing R1 for both, but R1 requires a separate ODAC download to get the 64 bit OLE DB driver.

One thing to add: the R1 installer lets you rename your HOME, while R2 doesn't. If you don't rename HOME_1 to something else during the R1 installation, R2 uses HOME_1 by default and will overwrite the existing folder groups in your Start menu.