OJS (and other 3.x releases of PKP software) supports separating the publication of content from the backend workflow. See: Settings → Distribution → Access.
This is primarily used when OJS is being used just for the workflow, and the reviewed and proofed galleys will be manually posted elsewhere. Others do, however, utilize the API to present the published content in a front end UI outside of the OJS reader interface. One intent of the API is to allow this to be done dynamically or in periodic tooling, which would facilitate what you describe.
Another approach would be use a purpose-built tool overtop of OJS, such as in-place web server caching, or, such as web scraping and caching, to improve native performance or to create a static copy. Such tools are independent of target site architecture. For scholarly publishing in particular, though, note that in the event of a site shutting down, this is exactly what LOCKSS does.
The scenario of hosting both a PHP-enabled server and a static copy of the site is, I think, fairly niche, but it sounds like there is some existing momentum around it. Please do continue the conversation here in sharing what you have already done. We’re happy to help connect any potential contributors with guidance within the PKP framework, and I think any gaps within the API for making this possible would be of interest to core PKP developers.