DPSS can be quite picky about their temperature of operation, but you can solder onto the pump diode like you would on any other.
If you operate them at too high or too low a temperature what will result in a lower amount of green light produced, but this does not damage the system in any way - just wait for it to cool down and it will work as it did before.
One thing that is possible is that you soldered it so hot the driver got damaged one way or another and now doesn't run properly anymore.
It's not very likely though if you used a soldering iron intended for electronics work and didn't keep heating it until the whole thing was hot enough to melt solder onto. Most electronic components can take quite a thermal beating when powered off. The spec sheet will tell you the minimum abuse it must be able to survive, but practically it often is much higher: I've seen plenty of transistor still work even after clinging to the soldering iron for several seconds adhered to those by molten lead-tin solder.