This is actually on the wrong board here on Rootschat.
You may find that the board moderator moves it
You don't mention David's occupation.
Do you know it?
Have you searched this site for the surname?
https://www.londonlives.org/index.jspThere are a few hits for David Wait which may or may not be him.
There is also this site
https://www.londonroll.org/homeIf either he or his possible brother James took an apprenticeship for their job, they may be in these records.
Unfortunately, the Coopers' Company does not seem to be included in the ones on this site.
However, it's always a good idea to search all the companies, as I have found they didn't all take an apprenticeship within the company you might expect. For example, I have an ancestor who had an apprenticeship to be a dyer, but it was with another dyer who was a member of the Drapers' guild, and not the Dyers' Company (I went on quite a wild goose chase for some time with that one!).
Here's the site of the Coopers' Company.
https://cooperscompany.co.uk/about-us/company-historyI see findmypast have some apprenticeship records, but I don't have a sub, and they may just be the records from the londonroll website.
Have you traced David's wife Eleanor Rice any further?
If you have found her father, have you looked for a will for him when he died?
No doubt you have realised this, but I would also search under 'Waite' and even 'Watt'.
Added:
Are these David's children?
David born 1774;
Joseph Thomas born 1776;
Christian born 1779;
James born 1780
All at Shoreditch.
They may help with the name of his father if naming conventions have been followed.
It looks like James was a family name.