Installing combo switch
Yes, but you're not going to like it, as it involves a hunt for high-tech products which may not yet exist.   We are not a product-reco site, so we can't be a whole lot of help there. 


Currently the white and black in the switch loop are always-hot, and switched-hot.   Re-designate them always-hot, and neutral.  
The hard part.  Find a "smart switch" product which allows you to have a master control at the switch location, and a remote module up in the lamp housing.  


Since there are no spare wires, the control and remote will need to communicate via either a) data-over-power-line, or b) a radio-based scheme like WiFi, bluetooth, or some proprietary method.

If you have a 1-gang box at the switch, you'll have a real trick - trying to obtain a module which is both a smart-switch-controller and also a convenience outlet.  If that wasn't complicated enough, since it's a bathroom, the convenience outlet must be behind a GFCI protector - either in an upstream receptacle, or the breaker in the service panel. 

The switch does not need to be wired.  It could be battery powered and glued to the wall next to the old switch location.  Then you can use a common duplex receptacle in the old switch box and done. 

Or, give up on a manual switch, and go with a motion sensor up in the lamp housing.  That can be done with common off-the-shelf components.  You may need to check if this is Code legal.  Commercial bathrooms do it, though.