This probably goes againt the conventional wisdom, but in trying it, you have little to loose...
Take the oring out, and when reinstalling the forearm, stop tightening the screw when it hits the wood. Now, when shooting the gun, rest it on the stock between the screw and frame. This has worked very well for me with 4 Handirifles.
Next is understanding the length of throat and how that may affect your accuracy. Obviously, the longer the throat, the greater the jump the bullet has to make untill it is suppoted by the bore. This can lead to some terrible accuracy.
First, how far out can you seat a bullet and it touch the lands? It is not uncommon to find that a bullet will not touch the lands untill it is out of the case in some Handi barrels. Begin, by sizing a case, and then lightly seating a bullet, seating it deeper each try untill the action closes. Deeper is not more than one turn in on the seating stem. Once the action will close on a bullet, it is time to begin load development.
Choose a midrange loading of several powders and fire them for accuracy. Take the most promising, and either raise and lower the powder charge from there and or change the seating depth. NEVER seat shorter than is reccomended with a given bullet, this reduces the case volume and increases the pressure. Work with these variables untill the best powder charge/bullet/seating depth is found for each combination.
THIS IS SLOW, PAINSTAKING WORK. No it ain't easy, but can be very rewarding.
The powders you are using strike me as a little slow for the 7mm-08 case. You might want to try out something a bit faster like 4895 or BL-C2, H335 somewhere in there. The powders you list are a bit better suited to cases with the volume of the 30-06 as compared to the 308. 4064 is a great powder, it will make any bottleneck case go bang, it may not deliver the best accuracy your barrel is capable of. It is in the right burning speed range though. Reloder 15 is about as slow a powder you want to work with in the 7-08.
Also, based on observation of what people say is working for them in Handi's, there is an overall trend toward the lighter bullets for caliber. You may want to try some 120's and some 150's as well to see if your barrel leans toward a given weight.
Directly inanswer to your question, it has taken at least 200 rounds before my guns became relaibly consistent.