Start with your GP/internist and tell her you think you may have exercise induced asthma because you have difficulty breathing when you're riding - describe the symptoms, but leave out the hyperventilation speculation. Ask for a referral for a pulmonary function test.
It's possible you may have EIA and a "rescue inhaler" like albuterol may do the trick. But they should do a test and make sure there's nothing else going on. (Says the person who struggled for months with "asthma" only to find out she also had pulmonary emboli....)