How to deal with an unsupportive SME ?

There are so many limitations imposed by employers as to the correct course of action. If your employer's perception is affecting what you can do, put the problem on your employer's lap. Let them know the trouble you are facing, and have them use their clout.

The more the problem is "yours", however, claim the issue. I hope the project-completion deadline was contracted relative to the participation of the SME (if not, it might be better to scrap the project, return the money, take the loss, and start again with a better contract). Give the SME deadlines. They are subject matter experts, not gods. They are supposed to inform the project. Fill in the well documented blanks spots with Lorem Ipsum, then give them the documentation so that they can fill in the blanks. You can "offer" a project extension. "Let's move this project back until 2023 March", and have them sit on the issue.

A lot of that can come of as snide, and if you need a more compassionate approach, you can ask about their workload and let them vent. Then you can go above and beyond by posing solutions for their backlog, or suggesting that a subordinate could write up one of them well-documented missing sections. 

Its important to keep those good relations with SMEs, and that sometimes requires playing the emotional support role. We are teachers, after all, and at least in America that is kind of expected behavior. Learners don't learn if they have an emotional block; and SME's don't SME if they have an emotional block.

I hope that helps,

P