Old Southern Rock Guy,
If you see air bubbles being forced back through, then it's not a
matter of siphoning. I think the drain that serves these fixtures is
partially clogged, or undersized. Now that you shoot water down with a
garbage disposal, it can't drain quite fast enough, forcing trapped air
up through the downstairs sink.
Perhaps the vents were never right, or perhaps the bar sink was
installed at a later time, without the proper vent. It may have never
been noticed before, but now the added disposal is causing the trouble.
If you drain water without the disposal running, and there's no
problem, then the culprit is isolated. The solution might be more
difficult to figure out, though.