What is the best way to mount a LiPo battery in a project box?
If you want it to be permanent, I recommend VHB 4905 tape by 3M.  You can use it to secure the battery to an enclosure, then use it to seal the enclosure, then use it to attach the enclosure to something :)

It has a body thickness of 20-mil, so it's pretty compliant.  It's non-conductive (10^16 Ohms/square), and has a breakdown voltage of 630V per mil.

The datasheet for the entire line of VHB tapes is here.  Page 7 shows the electrical characteristics.  It's not cheap, but you can get some from Amazon for about $12 USD.

Don't use it if you want to remove the battery later. I spent a lot of time with safety glasses and an X-Acto knife cutting a LiPo free from VHB, and I'll never do it again. LiPo's shouldn't be bent, pierced, or punctured.

For a removeable solution, I like the Velcro idea mentioned by @Linkyyy in the comments.

As an aside, once I took some home and taped a pewter key hook to my wall. A year layer, when I tried to remove it, the paint and sheetrock tore away before the VHB! Nothing like a little quick spackling as you leave :)