How to attach signature

The Return of Tarzan, pub. 1915. First signature is, literally, hanging by a thread. The remaining signatures are reasonably intact and secure to each other via the original stitching. Several pages at end of first signature need some TLC and will need to be reincorporated into the signature. My question is primarily regarding attaching the “renewed” first signature back with the rest of the text block. My plan is:

1. Cut first signature from existing text block
2. Repair loose/abused edge pages, use Sekishu to, in essence, recreate/rebuild the signature
3. New mull across spine (wide enough to reach onto boards ON TOP of existing endsheets)
4. Sewing “new” first signature into place, through the mull
5. Because the case is a bit fragile, the lining and across the interior hinges was reinforced with Sekishu. The intention is to add new endsheets, sewn through the new mull then pasted down. This seems safer on the rather fragile existing endsheets/boards.

Subject to your comments/input, this does not do a lot to attach that first signature to the existing text block as all other signatures are attached….???? Is this a reasonable approach? Thank you for your comments.