Yes, you can replace the 808nm diode and the green laser wil be back online. The diode must be turned in the socket to alignm with the crystals. The beam will be the same.
Ýou need to tweak the driver too or it will outputting the same current and so the same laserbeam.
A better way is to buy the whole green module for a few dollars more.
I doubt the diode will screw out like in the pic. Most diodes are pressed in. The only modules I have seen with the threaded diode retainer is the old style NewWish modules. It will have to be pressed out, and the new one pressed in.
I have several Red and Green Modules removed from Laser Light Shows. All of them have a Screw in retainer for the Diode. Makes it easy to replace the diode.
As for upping the 808nm Diode power to increase Green Output power. Don't you then run the risk of Damaging the Frequency Doubling Xtals ?
I blew a 808nm in one of my 50mw Greens. I replaced it with the same power 808nm. 300mw.