Occasionally police depts. will accept applicants with medical conditions, but they have to be well-controlled for a certain period of time. This is to protect all parties involved: fellow officers, you, and the public. I'm sure you can imagine what would happen if you had an asthma attack at an inopportune time!
Also, I don't know how you would do during training, if you have to do the pepper spray exercise.
I Googled "police officer with asthma" and got several hits. I think it will vary by department and severity of the condition.
Dispatching is fun and exciting. As officers, we handled dispatch duties until our dept decided to civilianize dispatch. I think it is worth considering! There are also other civilian positions within the department, such as crime analyst, applicant processing, etc.
Having developed health problems at the age of 38, I understand how discouraging it can be to have limitations placed upon your aspirations. Try to keep a positive focus, and find ways to go around obstacles. It sounds like you are already doing that by thinking about dispatch. Best of luck to you!