A double phase single set trigger will allow the lock to be cocked to full or half cock without having to "set" or push the back of the trigger forward until it clicks. A single phase single set trigger must be set first to allow the sear enough play to let the tumbler go into half or full cock. Inletting the trigger and bar too close to the sear of the lock will not allow the lock to be cocked. A screw that goes into the trigger bar and first through a rectangular spring can adjust the spring tension. A set screw going upward through the trigger bar determines how close the bar on top of the trigger rests from the sear. Too tight and the lock does not cock. Barely too tight makes the set up a dangerous hair trigger.
Depending on the lock you may have to cock it to half cock before installing in the lock mortise. Some percussion hammers have a notch on its inner surface that catches the top of the bolster of the lock. Those without the notch depend on the nose of the hammer to be stopped on the nipple. If the lock when out of the stock is relaxed to where the cock is all the way forward, the bottom of the mainspring may extend beyond the contour of the lock plate. If so you will ruin the stock trying to force the lock into the mortise if not cocked, preferably half rather than full. When not being used it is always advisable to store with the hammer all the way down on the nipple and the set trigger not set. Your springs will thank you for it
From the picture of the lock with hammer removed there is a hole to the right and above the square stud of the tumbler (to which the square hole in the hammer is installed and held with a screw). If that hole is threaded it is usually to receive the lock bolt coming from the left side of a right hand lock. That there is no hole in the inlay to feed the lock bolt through is a mystery. A lock bolt going from the lock side into a hidden nut under the inlay would not work unless the hole on the lock is counter sunk for a head on the bolt. The bolt head would have to be flush with the lock plate in order for the hammer to fall. Single lock bolts in some locks can use a notched stud at the front that slides into a screw in the mortise. Someone already mentioned this. The screw with a flat inner head usually falls between the bottle of the barrel channel and where a ramrod hole or "tunnel" or channel is. I've built a flintlock pistol with this set up that used a single set trigger that was single phase. You certainly have an interesting and valuable historic artifact that warrant especial care. Good luck with reassembling it. A master gunsmith of muzzle loaders like Jack Brooks of Englewood, Co could give you good advice on it if he had detailed photos or better the actual pistol to examine. Depending on where in the U.S. you live I'm confident you can find master gunsmiths of black powder pistols. The Contemporary Longrifle Society has a list of many of them.
