Did you read my comment in your other thread, in which I explain why using the “Never” timeout option is the only way to waive the password/PIN requirement on browser restart?
I think the closest feasible approximation to what you’ve asked for here is a vault that behaves like the “Never” timeout (i.e., vault data will exist in an unprotected form on your device until you explicitly lock the vault), but with apps checking some kind of expiration timestamp whenever the app/browser is restarted, and then forcing the app to lock on restart if (and only if) the timestamp has passed.
Thus, the security of your data will depend on how often you restart the app or browser. For example, if close the app or browser at 12pm after having set a 4-hour timeout, then the key that decrypts the vault cache will remain unprotected until the first time that you restart the app or browser after 4pm (whether this is at 4:01pm or 3 weeks later).