You can buy any asset but it must be “wholly and exclusively” for business purposes for it to qualify for capital allowances. Any portion not used for business i.e.commuting should be disallowed from the tax computation.
That’s the case for most equipment,but bikes loaned to employees are exempt under the cycle to work scheme. Have a look here
https://www.gov.uk/government/publications/cycle-to-work-scheme-implementation-guidance
So just buy the bike, and claim it as an expense.
Edit: just realised you are a soke trader,not Ltd Co, so not technically an employee,so maybe you can’t benefit from the scheme,have a chat with your accountant
Excerpt from the hmrc guidance here
To help employees take advantage of this tax-free benefit, an employer can simply
buy a cycle and cyclists’ safety equipment and loan it to an employee for qualifying
journeys to work. This arrangement means that the employee’s normal salary
arrangements are not affected and is sometimes referred to as a ‘salary plus’
arrangement. It may be, however, that the employer wants to recover the cost of
providing the cycle and safety equipment loaned to the employee. Usually this would
be done through a salary sacrifice arrangement.