Your approach is creative, but it may not be the best option. Do a search for "island sink vent." You will come up with an island loop which is the traditional method of venting an island. An easier option is to use an air admittance valve (aav). You can run your 1.5" waste line right through to the rear of the sink base cabinet and attach your trap in a more traditional manner. Just do a search for air admittance valves and you should find all the information you need. Good luck.