''He owns a very popular bakery, but a problem threatens his reputation.''
- Não vejo problemas com esta sentença. Me parece natural/idiomática.
- Exemplos com trechos semelhantes, feitos por falantes nativos do inglês, mostram isso.
''Bender North, who owns the local hardware store, and Sep Howard, who owns a popular bakery, have been best friends since childhood.'' [Google - USA]
''Brett Favre's second return to Green Bay Sunday night was just the background to the scandal that threatens his reputation.'' [New York Daily News]