On the matter of testing the filament string, you could just put in one tube at a time, as this set is wired in parallel. Some cheaper sets were wired in series, which meant that just like a cheap set of christmas lights, when one bulb, or in this case, tube, goes, then the whole set would die. But in this set that is not such an issue. Also, I would like to emphasize, if you are going to try this, you must not only pull the rectifier tube, but clip all of the wires coming from the transformer leading to the 5z4 socket. Even if the tube is pulled, the socket will continue to arc, as it is faulty.
Looking at the 5z4 vs. 5y3: They pin out the same, but the 5z4 has a cathode, while the 5y3 does not. I'm not sure if this would cause a problem or not, but if I was in your position I would get a proper 5z4.
Here are the pinouts;
5z4 The cathode is attached to pin 8, along with the filament string:
http://www.nostalgiaair.org/tubes/5z4.htm
5y3 Note that the filaments do the work that the cathode was doing in the 5z4:
http://www.nostalgiaair.org/tubes/5y3.htm