I'm sure it's possible, but it wouldn't be easy.
The Inspire uses a "smart" battery with its own internal hardware for monitoring charge, balancing, and even automatically discharging to storage level in a set number of days if unused. Much of the software is relying on that hardware. There are a couple of signaling pins that connect along with +/- when the battery is plugged into the Inspire. I'm guessing these pins are a serial bus of some sort (DJI likes CAN bus -- maybe that's what it is). I could probe these pins and attempt a decode, but even with that info the next question is how to implement the functionality in another LiPo. Maybe the Inspire will work without it, but given the level of integration that DJI has here I doubt it.
Then you have the physical issue of fitting another battery to the Inspire. That's a general Inspire issue by the way -- it would be a challenge to attach anything to it. Not a huge deal, but more work.
DJI is certainly charging a premium for their batteries ($169 for 4500 mAh 6s), but in the grand scheme of things it's not a miserably high markup considering a typical owner will want just a few batteries (I have three at this point).
I am still liking the Inspire. It is like the flying equivalent of a pocket camera -- you take the photo with the camera you have. Just as I would love to be able to instantly access my Canon 1DX with a big heavy lens on it (as opposed to the RX100 in my pocket), I would like to have my big octo with a BMPCC or GH4 ready to go for aerials. That's not always easy. Just for example, I can throw a pelican case with the Inspire in it on the racks of my 4-wheeler and head out to remote areas on the farm -- through dust, mud, etc., -- for aerials with ease. Getting my big octo anywhere is not nearly as easy.
So, for me anyway, this Inspire is all about convenience. It packs easily, travels easily, has minimal prep involved pre-flight, and there is nothing about it I need to change to make it work the way I want. It's the best out-of-the-box experience I've ever had with a multirotor.
The downside is complete lock-in to DJI. If this thing breaks it's almost certainly going to DJI service for repair. I'm a fix-it-yourself guy and that's not really on the table with the Inspire. And, as I've mentioned before, the camera is just fine for the purpose, but no GH4 or BMPCC. For a lot of uses that's no big deal, but just a limitation to manage.