Easy. First, start a conversation, and get it in writing. This gives you a time marker for how long you've been waiting, and gives an undeniable statement of work expected.
Hi SME. Excited to have the chance to work with you. Here is the list of documents that I need you to review: 1,2, 3, etc. Usually this takes about XX minutes per document, so the whole set should take around Y hours. Due to [externality], I would like to have these in the control system by no later than this [date].
Chances are he'll complain. That's fine. Complainers gotta complain. When he responds with a list of gripes, be graceful but firm. Simply say:
Hi SME. I understand your concerns, thank you for sharing them with me. You are concerned that review step X takes too long, and that comment step Y is inconvenient. I agree that the process is not perfect. However, the rest of the team has their documents completed using the current system, and for the present I do not wish to change the procedure that we have already started. Please complete your documents by [date].
After a few rounds using the system as outlined, we will have a review meeting and talk about what works well, and what needs changing. It'd be great to have you attend that meeting so we can get your input.
If the date above is impossible, please let me know promptly so that I can ask [his manager's name] to assign someone else from [field of expertise].
After that, it should just take the occasional reminder. As the deadline approaches, if he is still avoiding and complaining, perhaps send a more direct reminder and CC his boss -- preferably on the same email thread as your previous communications so the boss can see you've been asking for a while.
Message: Hi SME, I noticed today that you haven't completed any documents. I realize that you are very busy, so please let me know if we need to get someone else assigned.
Best regards, Myles
It's not your job to win SME over, or make him a champion of the process. At the end of the day, you just need the work done, or for his manager to assign someone else. Get it in writing, be polite, and don't involve his boss until it is apparent that he cannot or will not meet the deadline. Chances are, once he does it a few times, he will realize it's not so bad -- or maybe he'll suggest some genuinely helpful improvements on the next committee meeting. But at this point, you just need to get him (or someone) started.