I'm guessing not, unfortunately, because the $20/month is only supposed to be used for bike related things and maintenance.
From the IRS:
Qualified bicycle commuting reimbursement. For any calendar year, the exclusion for qualified bicycle commuting reimbursement includes any employer reimbursement during the 15-month period beginning with the first day of the calendar year for reasonable expenses incurred by the employee during the calendar year.
Reasonable expenses include:
•The purchase of a bicycle, and
•Bicycle improvements, repair, and storage.