I'vve never done this but have thought about it several times. Here is the plan I came up with.
Boil 5-6 gallons of water for a good 15 minutes and transfer it while still quite hot to a purged corney keg. Close it up, attach to pressure and let it cool. The pressure keeps it from losing seal as the volume shrinks during cooling. Then do a closed transfer from the imperial stout to a third well purged keg (the blending keg). Well purged is key because this keg will probably only get 10-20% full, I'd do a full volume star san push to get as much O2 our as possible. Mix by weight. If you want 80% imperial + 20% water transfer 8 pounds of the stout to the blending keg and top that up with 2 pounds of water. A reasonably accurate digital bathroom scale with 1/10s of pounds should be sufficient if you don't have a big postage scale.
Force carb in the keg and bottle with beer gun or counter pressure filler. With enough care in avoiding oxygen exposure you may be able to provide a beer with enough shelf life for a competition.