What pitfalls exist with running FastQC on a bam file?
I don't see why you are trying to compare two totally different measurements.
When FASTQC says a read is unique, it means that no other read shares its sequence.
When STAR says a read is uniquely mapped, that means it maps to one and only place.  That does not mean that there aren't a hundred more identical reads that map nicely to the exact same place.
You could have a very high level of PCR duplication of reads that nicely align to one and only one place. Or you could have a whole bunch of reads that differ slightly from each other who all map to 5 locations each.
FASTQC was designed for DNA sequencing, not RNASeq.  Some of its warnings make no sense in the context of RNASeq.  If I was doing RNASeq of a filtered set of pancreatic alpha cells,  I would expect a hell of a lot of reads for glucagon, and a lot of them would be natural duplicates, and that would be biologically normal, and not a problem.
You could take the same batch of reads, and align them to one genome, and align them to a mix of two closely related genomes.  The second will have far more non-uniquely mapped reads, but the FASTQC stats will not have changed.
That said, if you doing human or mouse RNASeq, 1% unique alignment means something is wrong.  Find out where all those reads are aligning, you might have rRNA contamination.  If you are doing an organism with a high level of duplication in its genome that might be normal.