Actually, from what I've read and heard elsewhere, "The Lost King" is based on the real life efforts of Phillipa Langley in uncovering the remains of Richard III. The "Society" has been around forever and their involvement seems to be marginal if at all. Other parties and persons were much more supportive of Langley in what she undertook. Most of the news links describing the film provide the wider sources of who participated and assisted Langley, or who took an interest once the issue of the reburial arose.
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The website of the Richard III Society has a great deal of good information about the discovery and people responsible. Worth a look.
https://richardiii.net