After further testing, I realized it was not the end-all issue. On my first computer I was testing, I had the C:\Program Files (x86)\Mozilla Firefox folder in my %PATH% variable. Therefore, when PDQ executes Firefox Setup 46.0.exe, Firefox 32-bit would try to interpret the command, and just kind of... Stall? I removed the variable, rebooted, and Firefox deployed to that first computer as well.
Might be a bug with other programs, but I chatted with Jason via email, explaining what may be a good trick for avoiding that in future builds. His support was pretty awesome during all this.