You don’t pay for a Duty solicitor, no one does, it’s your right.
You can bring in your own , paid , solicitor if you want and think they would do a better job than the duty solicitor.
Legal aid is means tested.
If you can afford a solicitor you can’t get legal aid.
If you cant afford a solicitor, you can get legal aid.
Why you are conflating duty solicitor with legal aid, two different things , is beyond me !
Cheers.