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BBSA 8.1 supports 2008 R2.
Review the following KB for your error message:
KA324245 - BBSA Windows Patch Deploy Job error: BlPatchCheck2.exe failed. Return Code: 1
Following the knowledge base article did the trick. I re-registered the BlPatchCheck2.exe DLLs and BBSA was able to install most of the required patches on my Windows 2008 R2 server. There were a few failures with individual patches that I'll have to investigate but it's working much better than before.