If your friend has a touchstone back he won't want a new battery door. That would negate using the touchstone.
The Mugen 1400 is a solid value. It really adds power and longevity of use to a Pre, Pre Plus or Pre 2.
I LOVE mine and am soooo happy I got one. It fits tight under the existing back and doesn't spoil the form factor of the phone..
Does your friend have Dr. Battery? If the Health of the battery is low (near 80%) it is past time for a new battery anyway... Even a cheap battery, if new (and IF it is as advertised), would give better performance and a Mugen 1400 would be a huge boost.
And perhaps some tweaks are in order to maximize power conservation...
Lumia 1520.3 (the Beastly Unicorn): Windows 10 Mobile
Using an extended battery I got for 5 bucks on craigslist the Doc showed it to be 98% life....it lasts all day with super heavy use....came with the elephant back....Games video constant web use...and hours of texas hold'em....still at 84% today