The best you're going to do with a freewheel and brifters is a 7-speed Shimano freewheel and Shimano RSX brifters. Granted, I'm talking theoretically, as I've yet to do the setup myself - but - the spacing on a 7-speed freewheels and 7-speed cassette are identical, if my memory is still working. You'd need to change over the derailleurs to RSX or something Shimano that's newer (there are a couple of exceptions, mainly old Dura Ace).

I'd seriously suggest digging up a 14-28 6-speed freewheel (I assume what you currently have is narrower), and drop your smaller chain ring a couple of teeth. Short of tree climbing with loaded panniers, that should do for general use.