I do not believe it is possible to pass custom variables via the URL to open the base.
You could however host a custom page (on your own server) and embed (via an iframe) the base on said page.
Since the page would be hosted on your own server you could thus pass any custom parameters and read the same
However point to note you will not be able to change formulas etc in the embedded base via the passed parameter as Airtable does not allow that via their “iframe” functionality.
I know this is not the answer you were looking for but hopefully it gets you to start thinking of some other work arounds.
All the best!