Since you're not dealing with a limited company, the law regards 'Fred Bloggs', 'Fred Bloggs (t/a Bodgit Carpet Fitters)' and simply 'Bodgit Carpet Fitters' as the same entity.
Indeed the law requires that any business documents (such as invoices) issued by Bodgit Carpet Fitters MUST show Fred Blogg's own name and address (or an alternative address where legal documents can be served upon the business).
So your letter should be addressed to Fred Bloggs, rather than to Bodgit Carpet Fitters.
He won't be able to get out of paying up unless he goes
personally bankrupt, since (as I've indicated above) his finances and those of his business are the same.
Your letter needs to include . . .
1. Your name and address (obviously!) ;
2. The name and address of 'Fred Bloggs' ;
3. The date upon which the letter is written ;
4. A statement of the amount demanded (or action which is acceptable in lieu of payment) ;
5. The reason for that demand ;
6. Notification that if payment isn't received (or remedial action completed) within a specified period (I suggest 21 days) that legal proceedings will be commenced without further notice.
To avoid any possibility of Fred Bloggs telling the County Court that he never received your letter, it's wisest to send it by Royal Mail's 'Signed For' service, even if Fred Bloggs only lives next door to you.
If your letter doesn't result in the matter being resolved, you'll probably need this web link:
https://www.moneyclaim.gov.uk/web/mcol/welcome