Have WordPress generate a JSON of the content

After pondering your question some I am guessing you mean to generate actual physical JSON file in filesystem to use as data source? That would certainly be unorthodox in WP development.

Typically in WP you try to minimize disk access, since it is more likely to be a bottleneck in general. Doing file writes is quite inconvenient in WP extension meant for public distribution, but in private of specific server you could generate and dump JSON into a file rather trivially.

For a more “WordPress way” of doing things you might want to look into persistent Object Cache. WP can be made to use persistent memory cache (such as APC, Memcache, Redis, etc), which both greatly increases its general performance and makes it possible to cache data there via WP APIs.