I had initially wanted to use the 3 volts the original battery holder provides with 2 AAA batteries. I could not get the circuit to work with only 3 volts. So I decided to use the Adafruit 150mAh LiPo battery. To connect the battery, I removed the AAA battery tab on the upper right and drilled a hole. I used a JST extension cable to connect the Sound FX Board (via the JST connector soldered onto its backside) through the hole to the battery compartment.
The 150 mAh battery fits in the battery compartment after breaking a bit of the plastic that separates the two AAA batteries.
You will want a LiPo battery charger handy to top-up the battery. If you plan to use the prop for a long stretch, a larger battery may be better or a spare battery to swap. While I did not have an off/on switch, adding that would help save battery power.
Solder the speaker wires to the amplifier outputs. I used the mono output from the left side which worked well.
Place the electronics back into the body of the prop with the trigger board placed back in the groove it was pulled from. Feed the battery extension through the hole and attach it to the Sound FX Board. Be sure the trigger assembly is reconnected to its spring (if it came loose) and that it presses the silver blister on the switch board. Connect the battery before assembly, to be sure it works. If not, pull the electronics out for retest. If working, get the board stowed into the prop.
For plastic reassembly, insert the rear plastic piece under the barrel rear (is this some sort of hammer? I don't know), and place the screws in to hold the halves together.
Place the parts you removed with the heat gun (end cap, barrel cap, two straps) back on the barrel assembly. They may snap on or you might have to use glue. If they were misshapen during removal, you might use a dremel to shave the plastic a bit.
If it works, go have fun with your new prop!