Possible? Yes.
Practical? Not very...
Rebarreling to .300 Whisper would be pretty simple. A bit more difficult than replacing the 5.56mm barrel with a new 5.56mm barrel, since you'd have to find someone willing and able to make the barrel. Fluting the chamber would be the most difficult part: there are plenty of smiths who could make a barrel that would fit an HK91 trunion and front sight assembly, but not so many who can flute the chamber.
If you were going to suppress the rifle---and that's pretty much the only reason to build a .300 Whisper rifle---anyone who could make a .300 Whisper barrel for you could probably do the porting and/or threading for a suppressor. An integral suppressor would seem the best plan, since a muzzle-mounted unit would add a lot of bulk to the weapon in comparison, and you'd already be dealing with having a barrel custom-made.
I don't know if it would be necessary to modify/replace the locking piece. I suspect it might be, since H&K found it necessary to design a new locking piece for the suppressed versions of the MP5. You'd need to find someone familiar with the design who could fabricate a new locking piece, and this isn't likely to be cheap or simple to do.
But even after all of that trouble and expense, an HK93 wouldn't be a very practical vehicle for a .300 Whisper rifle. You would have to be willing to dedicate an HK93 to this use, rather than being able to simply switch upper receiver assemblies as is possible with the AR15 family. And the HK93 is pretty rough on cartridge cases. As .300 Whisper is an almost purely handloaded cartridge this would drive up cost and effort considerably.
An AR15-based .300 Whisper would be much simpler if you were looking for a semiautomatic rifle. For the quietest possible rifle there are small bolt actions that would make good vehicles.