With the 9V 650mA hooked to the Uno and the nte:The 9v 650mA was loaded just enough to keep the nte and your crt. happy (you got lucky).
With the 9V 650mA hooked to the nte only or the 12V 7Ah battery to nte:Nte not happy, too much input voltage to output 5V @ >500mA
To get the rated 1Amp from the nte you need:A heat sink, not much more than 7V on its input and perhaps a small sacrifice to the electron god.The more voltage you input to the nte the less current it can provide.
The spec sheet rates this regulator at 35V max input voltage and "Output Current in Excess of of 1.0 Ampere".
Don't need the whole amp, just .7 or so. And would the the 2 extra volts over its minimum operating voltage cause that much of a problem?
When I got everything working 100% I wasn't using the the input/output caps or bypass caps, so I didn't include them here.
Ok I placed a 22 pF ceramic capacitor on both sides of the dip socket across +5v and gnd. Both capacitors are close enough to touch the socket.