You are correct about damage caused by dry firing. The PIAT is just like compound bow. Neither can be dry fired and both have to be shot with their specific projectiles. In the case of the bow, you can have the whole thing just self-destruct on you. Bow strings can snap and the bow limbs can shatter -- this is all very dangerous to the shooter. In the case of the PIAT, you've explained the kind of damage the previous ownder did by his fooling around with it.
As far as a representative type or types of projectiles go, you may just have to rig up a dummy round of the type. The PIAT projectile was not very sophisticated and it should be relatively easy to replicate in appearance, size, and approximate weight of the HEAT (high-explosive, anti-tank) warhead. (I think the only other type was a practice round.)
As I recall, and please correct me if not, the PIAT was a spigot mortar similar to the Hedgehog anti-submarine weapon. (Hedgehog resembles an overgrown U.S. M9A1 rifle grenade.) The Hedgehog had an impulse charge in the tail boom. The hollow tail boom slid over the launching spigot that was one of 24 in the launcher that was bolted or welded to the ship's structure. The firing pin in the top of the spigot rested on the primer of the Hedgehog. When firing voltage was applied to the primer by the firing pin, the primer fired the impulse charge and sent the round flying towards the target. Recoil from the departing Hedgehogs (there were 24 that were ripple-fired in pairs at 0.2-second intervals) was considerable and that meant only larger ships like corvettes, frigates, or destoryer escorts and destroyers could carry them.
In the case of the PIAT, the launcher was cocked manually for the first shot (quite a feat as I recall), and the PIAT anti-tank bomb's tail boom assembly was slid over the spigot in the launcher. Like the Hedgehog, the PIAT bomb had a primer and impulse charge in the tail boom. When the trigger was pushed, it released the spring-loaded spigot to hit the primer and fire the impulse charge. The resulting recoil caused by the departing PIAT bomb re-cocked the spigot for the next shot. Memory says that users reported the PIAT recoil was not pleasant. Also, if the recoil did not re-cock the spigot, the PIAT gunner was in a lot of trouble because he'd have to do the job manually and with considerable effort. Re-cocking a PIAT under fire was not for the faint of heart!
The WW2 U.S. contemporay of the PIAT was the 2.36-inch (60mm) rocket launcher (Bazooka), M1-series or M9-series or M18. The Bazooka fired a fin-stabilized rocket that carried a HEAT warhead for use against armor. There were also WP Smoke and practice rounds for it. Unlike the PIAT, the Bazooka operator didn't have the recoil to contend with. The Bazooka's operator had electrical problems to worry about. The first Bazookas used dry cell batteries to fire the rockets electrically. Operators always had to worry about whether the battery was good until the firing mechanism was changed to a magneto. The magneto solved one of the major problems of the first Bazookas. The firing conduit to the contact studs for the rocket firing wires was vulnerable, but the rate of fire was slowed by the need to wrap the bare firing wires of the rocket around the contact studs. This could be a nervous task under fire.
Late versions of the Bazooka had a contactor latch that applied firing voltage directly to the fin shroud of the rocket to fire the motor. All rockets retained had the original firing wires in case the rocket had to be used in an earlier version of the Bazooka without the contactor latch. The combination of magneto firing and the contactor latch made the Bazooka a pretty simple and effective weapon. It was light in weight and could function both as an anti-tank and anti-personnel tool for the infantry. The PIAT was never as extensively developed as the Bazooka rocket launcher. Korean War versions of the Bazooka were increased the bore to fire 3.5-inch (88mm) rockets. The larger rockets were needed to carry heavier warheads to defeat tanks like the Soviet T-34. The PIAT was hopelessly out-classed by that time.