First, the device you are thinking of, is an air admittance valve.
They are one way air valves, that let air in, but not out . They are
used in areas where a conventional through the roof vent is not
possible, like your island kitchen example. However, they function
along with a regular trap and as far as I know, come in sizes made to
fit that type of arrangement, ie 1 1/4 up. So, it not clear how you
can easily work one into your arrangement.

Second, are you sure that is your problem? What I'm having difficulty
with is how a syphon could continue to operate. I would think that as
soon as the dishwasher is pumped out and there is no water left, the
syphon would break by drawing air into the line from the bottom of the
dishwasher. That should happen before it starts to fill again.