Solder up the wires to your trinket according to the circuit diagram, shortening them considerably.
Solder a JST battery connector to the back of your Trinket.
Now when you program the circuit over USB, you'll notice that the motor and laser don't power on. That's because they are wired to the battery power, so any changes in your code will only be visible after you disconnect USB and reconnect a battery.
You can either solder the switch directly to your battery or use a JST extension cable to make a switch adapter to go between the battery and Trinket.
Stick the Trinket to the bottom side of the servo and gaff tape the battery to the outside flat side of the servo. Try your new laser dog goggles on and notice the angle of the assembly and how it affects where the laser points. Make any desired adjustments by removing the motor-laser assembly and re-hot-glueing it on at a different angle.
After you've found your ideal laser angle, use E6000 adhesive to permanently attach the assembly to the side of the dog goggles, tape or prop up in place if necessary, and let dry for at least 24 hours. Remember adhesive give off fumes and shouldn't be placed near your pet's face until completely dry.