autofill only working directly after DB unlock

### Checks

- [x] I have read the Wiki, searched the open issues, and still think this is a new bug.

### Explain the problem clearly and succinctly:

### Error when:
0. make sure keepass DB is unlocked (ie. used shortly before)
1. open Firefox (149.0.2) and navigate to a webpage for which credentials are stored in keepass DB
2. click into i.e. username inputbox
3. select correct entry
--> login data is NOT auto-filled

### SUCCESS when:
0. make sure keepass DB is locked
1. open Firefox (149.0.2) and navigate to a webpage for which credentials are stored in keepass DB
2. click into i.e. username inputbox
3. click on "Sign in with KeePassDX"
3b. use biometrics to unlock DB
4. select correct entry
--> login data is correctly auto-filled

### Describe what you expected to happen:

autofill should work, regardless if DB is already unlocked

### KeePassDX version:

4.3.3

### Build:

Libre

### Database version:

_No response_

### File provider (`content://` URI)

_No response_

### Android version:

16

### Android device:

Samsung A55

### Additional context:

_No response_

Reproduce the reported steps in Firefox on Android 16 with the KeePassDX database already unlocked, then compare them with the biometric-unlock flow. Trace the autofill entry point and determine why selection fills credentials only after unlocking; done means autofill works both when the database starts locked and when it is already unlocked.

Written by the indexing model from the issue text.