Two possibilities: I'ts the gun or it's the brass.
Since you are using mixed headstamp brass, it seems possible that some pieces have deeper primer holes than others. Check visually to see if the problem is restricted to one brand of brass, or if the primer seems deeper in the hole on brass that did not ignite. Since fired primers move in the hole due to pressure, it will be necessary to reseat the spent primers before making this comparison.
Another, even more likely possibility is that the brass which misfires is too short. The 45 ACP round headspaces on the case mouth, not on the rim. Look into the cylinder chambers of your 625, and you will see a ridge in each chamber. The case mouth of each cartridge should rest against that ridge, the same as it does in a 45 automatic. If the case is below specs, some of the energy from the firing pin will be absorbed in movement of the entire case forward until it hits that ridge. So less of the impact goes into the primer. There are specifications for case length which you can use, but also compare the misfired cases to ones which did fire. You will need to resize the fired cases to get a valid comparison.
There are several ways to lighten trigger pull. One is to "adjust" the strain screw by unscrewing it a bit. This is not a reliable method since it can continue to turn during recoil, but that doesn't stop some folks. Check it to see if it is tight. Another method involving the strain screw is to shorten the tip until the desired trigger pull is achieved. This is the method that the factory uses (they are looking for a certain amount of resistance in pounds), and I use it as well to lighten trigger pull past factory specs. If you shorten it too much, you will get unreliable ignition.
A third method is to bend the claw end of the mainspring to de-tension it. This is a sloppy, unreliable, and unrepeatable method but some folks use it anyhow. Compare your mainspring to a new one or one from a known good gun to see if the shape is different.
My S&W revolvers all have very light single action pulls, even as they came from the factory. You barely have to move the trigger before the sear is released. This is normal.
I will not advise you to go out and buy different firing pins or other things until you have sorted through these possibilities. S&W revolvers normally do not need aftermarket parts to function properly.