When you say a "bench rest rifle", do you mean a rifle that you plan on shooting from a bench, for your own satisfaction and fun? Or do you mean a rifle to use in formal benchrest competitions (e.g. IBS, NBRSA)?
If the former (just a good accurate fun rifle), you can do a fair bit with a 788.
If the latter, a 788 would be completely inappropriate. It would be far cheaper, in time and money, to buy a $1000-$1500 custom action, and have built for you a $3000-$4500 custom BR rifle. If you're looking for a cost-effective way to get a competitive BR rifle, accurizing a hunting rifle isn't it. If you're a stubborn gunsmith who is determined to prove your abilities and show that you are able to make a hunting rifle action shoot as well as a custom action, I suppose this is as good a windmill to tilt at as any.
Remember that if you are getting a good custom rifle built, the action is only one part of the overall cost. A good barrel will cost you $X, whether it is mounted on a "free" 788 action that you have, or if it is mounted on a $1200 custom action. A good stock will cost you $Y, regardless of what barreled action you install in it. A good scope will cost you $Z, regardless of what you mount it on. A good benchrest and bag will cost you $Q, regardless of what you shoot on it. Your reloading gear setup will cost you $R, regardless of of your rifle.
There is one difference though. Your gunsmithing on a custom action will cost you a very small fraction of what it will cost to get a factory hunting rifle action (e.g. Rem 788, Rem 700 etc) tuned and accurized to BR levels of accuracy. It _can_ be done, most of the time, if the factory action isn't too badly flawed, but it is rarely cost effective. It will almost certainly be the case that a "free" factory action plus the required accurizing, will end up costing more dollars than a brand new $1500 purpose-built action.