Find out CPU usage by reporting server
Regarding running reports from an Availability Group secondary server, you will need to license that instance of SQL Server.  Since the AG requirements require identical equipment, your secondary's licensing cost will reflect the primary server's licensing cost

Using the secondary server strictly for High Availability/Disaster Recovery does not require licensing the secondary server since it is only in use when the primary fails.

Regarding the brief spikes in CPU utilization, is it causing you a problem now?  Since it is hard for you to see who is responsible, consider loading sp_whoisactive from Adam Machanic's site: http://whoisactive.com  This stored procedure has numerous options to control what information you can query.

Brent Ozar also has an article on using sp_whoisactive to track issues within your system, logging the output to a file for later review.  http://www.brentozar.com/responder/log-sp_whoisactive-to-a-table/

Perhaps these tools will help you find out what is happening.