How do you deal with functional managers in a matrix organization?

Assuming the sponsor has some weight in the organization, they could assist. The weaker the organization (and this is a matrix so 50/50), the more support you might need from sponsors and influential stakeholders.

I would take it up with the person the functional manager reports too. The manager sound like he/she has no confidence in the resources working on the project. The manager is trying to protect his/her staff.

Cross matrix team challenge!
1.Understand what 'unrealistic' means for the functional manager. Is it the resources number/project duration?
2.Negotiate to get a win-win situation for company and project.
3. Backing of upper management. It's not 1 functional manager. It involves the company's stakeholders which are backing the project. (Get all stake holders involved which are involved in this project plan. Is this is what only the team worked to come up with project plan , there must be some input from upper management regarding the scope and work in the plan. Reduce the scope if all are not agreeing to project plan and get approvals.)

Agree with Sante support of influential stakeholders and sponsors are need to get support of Functional manager. Moreover alignement of project to strategic objectives of organization is also likely to get support of all concerns

Does the functional manager mentioned into your story some influence? Two activities, focused on project, must be done from the very beginning, both closely related: elicitation and stakeholder management. If you performed both and you face this situation (by the way, I am facing this type of situations often including top management not functional management only) then it is key to answer this question: who owns the project? Project manager and project team do not own the project. So, if the functional manager owns the project then work with them and all related information to get the project plan the functional manger thinks it could be. If the functional manager sill insist into other plan than the plan the team got following the defined process then you have to request the approval. If approved as functional manager desires then record a risk and assign the risk to the functional manager.

Agree with above comments, but like to add a further thought: Assuming the project the complex and highly adaptive, to come up with a "realistic" upfront plan is in most cases not realistic and even pushing the team into certain wishful numbers by management is from my point even unethical. The one who are supposed to execute the project work are the ones who know best what is needed to do. For the first iterations, they should be able to come up with robust figures, for the further future it will be still uncertain and a forecast. The team needs to learn over the project and the estimates (plan) will be more reliable. Pushing the team by direct and control into number and even asking them to commit to it, is from my point of view a big mistake. To that extend, take the courage to "educate" the functional manager for the benefit of the team (I assume the team is motivated and has the skills and knowlege as a team to get the work done).