This could already be done with stuff I have around the house. Years ago I put together an emergency AC power kit for when winter storms knock out the electricity for hours. I have a re-chargeable portable and sealed 12v car battery, which I plug an inverter into to change DC to AC, so I can plug lamps (with low power CFL or LED light bulbs) into the inverter. We also can keep our cell phones charged too, although the 5v USB portable battery banks can take over that task now.
Anyway, I can take the battery, inverter, and Echo out to the patio and plug the Echo power plug into the inverter. Since AC from an inverter is not as smooth as AC from the wall, it could possibly cause static sounds to also be in the music. It would depend on how well the Echo power supply can smooth out the AC coming in. The Echo electronics don't use much power, but its speakers do since speakers are really specialized electric motors, and motors use juice. So battery needs to be up to the task, and a car battery is.
Then, I may also need "portable" wifi, because I may find my home wifi doesn't have the reach needed for a good signal.
So, since I don't care to move my Echo around (I have plenty of other gadgets I can use) I am not going to go to all this trouble to even check this out.
Nevertheless, if you come up with a large enough battery (power-wise) that can be plugged directly into the Echo, go for it! Good luck.