From my understanding, the "share on a website" only exports a static version of the workpad. Also when I go to "share on a website", it says my drop-down filter is incompatible with that feature, and I'd like to include the drop-down filter on my public dashboard.
This is correct; the shareable is not connected to Kibana at all and only contains the static data from the JSON file. This is also why elements which manipulate/re-query data (like a drop-down filter) are disabled, as it would require fetching new data from Elasticsearch.
If you need users to be able to query Elasticsearch, you'll need to look into embedding Kibana in an iframe. For example, you could create the Canvas workpad you want to embed, lock the controls to prevent editing, and view fullscreen. Then use that URL in your iframe so that the fullscreen view loads up by default:
<iframe src="https://my-kibana.com/app/canvas#/workpad/workpad-{id}/page/1?__fullscreen=true" frameborder="0" width="1500" height="800"></iframe>
One important thing to note is that you will need to set up a dedicated user/role for the iframe, to prevent visitors from having full access to everything in Kibana. To do this, you'd want to do something like create a role with read permissions on the index you need to query, and a spaces-level permission that only provides access to the Canvas app.
You can then assign a user to this role and set up a proxy to auto-authenticate the embedded iframe to the new user as described in this thread.
Technically a user would still be able to hit "escape" to back out of the workpad's full-screen mode, but if the security permissions are set carefully, they shouldn't be able to do much else.
Hope this helps point you in the right direction! Longer term, we are looking at ways to make it easier to embed pieces of Kibana externally, but in the meantime this is probably your best bet.