ELK 930 and SkybellHD

Are the two chimes identical? 
 
If you disconnect the first (closest) one, but leave the second one connected, does that one then work?  If so, that would suggest that it is indeed a power issue and higher current is required.
 
Do you have a multimeter with an amps scale?  If you do, you can measure how much current each chime requires by connecting the meter in series with the chime and the power supply. I.e. connect one lead of the meter to the power supply, and the other lead to the chime, so the current flows through the meter.
 
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