Do I have to make an offer within the budget set by the Hiring Manager?

No. Hiring Managers set their budget for each position, and you can make an offer at their budget amount or higher or lower. It's your choice. Maybe you have a slow month and want to make a low offer. Busy? Maybe make a higher offer. Local to the event? Maybe make a higher offer because you won't require airfare? Hiring Managers don't hire based solely based on price. They also care about your skills, experience, location and of course what you say in your pitch!