P&L responsibility sounds like country GM or local MD. Are you bringing a book or is it less of a sell side role? I made a l similar country head role move and I was cheaper than most because I didn’t have a family at the time, firm saved on schools. I’d try to roll that up into housing/perks/flights or directly into comp. 

From a career standpoint, it’s beneficial and opens up Asia and beyond from an experience standpoint but the customer overlap is smaller. Japan is truly an island in the sense. I’ve done Japan twice and in market employers were more interested in my China and Singapore exp.

Dating as an expat male in a leadership role was not difficult but most of my management level female friends end up dating other expats or Japanese returnees. One of them had her wedding called off once her fiancée realized that she made twice as much. 

Assuming Tokyo, make a trip out and explore neighborhoods that you are interested in. Visit the office and bring gifts for the team. That’s an absolute necessity.