My first recomendation is to leave the sink in front of the window. I would say the majority of houses being built have the sink in front of a window. May help resale also.

If you want to move it still. Go underneath the house ( I noticed the wood floors) and move the drain line to the location you need to bring in up into the wall cavity. Run the vent all the way up into the attic, then tie the new vent into the existing vent. Remove the old vent as it seems it would then be a dry vent which is a no go on an inspection. Ensure you put a clean out in because your vent will not be a straight shot out the roof

Use an air admitance valve only as a last resort.

Thanks gwilson. I'm moving the sink to maximize counter space, in the previous layout the kitchen had several small, useless, spaces between appliances and the sink. I also didn't have a useful workspace near the range, which was annoying. Regardless, the cabinets have already been ordered a few weeks ago, so I can't change it now.

I'll look into running a new stack into the attic. Unfortunately I finished drywalling the bathroom above before tearing out the kitchen, so I may have to undo all that work.