How to invoke Jinja processing in my card JS code?

No bundling is a JS specific thing. Basically the compile step equivalent for working with JS/TS for use on the browser.

You’d need to put some work there before you jump into HA specific quirks.

As for caching, you can mitigate most of it by just testing using your HA’s local IP in the browser. Do a cache purge on CloudFlare or similar after actual deployment and you’ll be fine. That’s a browser/proxy specific quirk, nothing to do with HA.