The following seems to work:
1) Create the following script:
#!/bin/zsh
CLANG_OUTFILE=${BB_DOC_PATH%%\.m}.out
clang -x objective-c -o $CLANG_OUTFILE $BB_DOC_PATH
$CLANG_OUTFILE
2) Save it (with extension .sh, although this might not be necessary) in the Scripts Folder (typically ~/Library/Application Support/BBEdit/Scripts)
3) Write and save (somewhere on disk) the Objective-C source code (I used '.m' suffix which the script will remove and replace with '.out')
4) With the saved source file as the active window in BBEdit, select the script from the script menu. The results should appear in the Unix Script Output.log window.