It's natural, but it is tentative. It is apologetic. It is begging the person to make an extra effort.
When you ask somebody "Could you see your way clear," it means that the way looks blocked. You are asking them to look hard for a way to meet your request... or part of your request.
"I'm sorry, we do not have any rooms available for April 8th."
"OK, but could you see your way clear to putting us on a waiting list, in case somebody cancels their reservation?"
"Would you go out on a date with me Thursday evening?"
"I don't think so, I sort of have a boyfriend."
"Well, maybe not a real date, but could you see your way clear to at least going out for coffee at lunchtime?"