Plumbing Vent - Loop Vent, air admittance valve or what.?

This will vary by your local plumbing code. I would say to run it horizontal (slight upward angle) thru the cabinets until you are past the bay window. Then into the wall and vent it properly. Either by going thru the roof or looping around until you can tie into the existing vent.

Remember that what you do now may catch up with you later when you want to sell the house. If your local city or a buyer inspects the work and finds out there is an improper/illegal vent it could cost you major $$$ to retrofit.

Also remember to put in clean out plugs on your drains and shut off valves on the supply lines.

Best of luck

If you feel like ripping open walls than you can run a new waste vent. If not, you can do what I did and buy ( I call it a cheater valve). Install it as high as you can in the back of your sink-base. It must be higher than the drain basin in the sink. They work just fine, and if your doing it yourself you can get away without an inspection. So there ya go, good luck.