Is there a way to determine the decompressed size of a .bz2 file?

As noted by others, bzip2 doesn't provide much information. But this technique works -- you will have to decompress the file, but you won't have to write the decompressed data to disk, which may be a "good enough" solution for you:

$ ls -l foo.bz2
-rw-r--r-- 1 ~quack ~quack 2364418 Jul  4 11:15 foo.bz2

$ bzcat foo.bz2 | wc -c         # bzcat decompresses to stdout, wc -c counts bytes
2928640                         # number of bytes of decompressed data

You can pipe that output into something else to give you a human-readable form:

$ ls -lh foo.bz2
-rw-r--r-- 1 quack quack 2.3M Jul  4 11:15 foo.bz2

$ bzcat foo.bz2 | wc -c | perl -lne 'printf("%.2fM\n", $_/1024/1024)'
2.79M