When the MP3 module is actually playing a song, it's stomping on the RF signal to the receiver! It works great to start the first sound, but if I want to change the sound from the transmitter, I have to be pretty close to the receiver to get it to work.
Well, there's a clue. More than likely the module uses a Class-D switching amplifier so when driving audio to the speaker you have 5V pulses radiating EMI courtesy of the huge antenna of the long speaker coil winding. I notice in your prototype photo that the speaker is conveniently placed next to the RF receiver to maximize the stomping. 
So first move the speaker and wiring away from the receiver. That might do something. If you have one of those ferrite cores/toroids to wrap you own inductors, place it near the module output and wrap the speaker wires thru it. If that shows improvement but not enough, google "class-d emi filter" or something like that and install an engineered L-C filter right on the module.
Yes, I'm imagining how easy the job is...but IMO the greater contribution to the hobby would be if this was not dependent on using an ERR module to gain remote control...which would leave out conventional, DCS, DCC, etc. users.
