I'm not sure an LOE is sufficient. CIC may look for more proof since they will need to be able to verify that you and the person with the work experience are one and the same. Typically your employer would have your legal name on record. Is there a way they can use your legal name in the employment letters (and perhaps include your preferred name in brackets or something like that)? Failing that - I think I would at a minimum include an affidavit from you regarding the different name.
Where are you from? Do your payslips and tax statements from the places where you have worked show a social insurance number or some sort of similar government tax / employment ID? This might be another way to address this.