The only "Approved" lightweight battery is the SJS-J-16 battery, commonly referred to as the "Odyssey". It is a sealed lead acid battery, so can be installed on its side, which is the way I'd mount it, UNDER the extended baggage compartment, with a simple hinged door for access to the battery. The certification regulations specify an "Approved" lead acid battery.....and that Odyssey battery is TSO'd, so "approved".
I'd also mount the ELT the same way, UNDER the extended baggage floor. Picture below of ELT. Could easily do the same with battery laid on its side.
Mounting those things BEHIND the extended baggage will make access to them a BITCH! Unless, of course, you install a small access door in the side of the fuselage....meaning cutting into that recently painted surface....and, structure.....the skin is structure.
The other "lightweight" battery is the Earth X ETX 900. I don't believe there's an STC for that battery, and Earth X is quite clear that NONE of their batteries are "approved" for use in "Certificated" airplanes, such as the 170. They are being used in a number of Experimental aircraft, and the ETX 900 was specifically designed for that application, with various protection, etc.
I'd be reluctant to mount a lithium battery inside a fuselage, though the ETX 900 has overboard vents in case of venting. And, in any case, that far aft....
That said, the ETX 900 is ten pounds lighter than the Odyssey.

MTV