Putting two parts of text block together

So here are pics of two sections of Bible text block as well as the spine view of the larger section. Both sections are quite in tact and solid. Now they need to be “put together”, new mull (probably) across the spine before recasing. The spine “liner” across both sections has NOT been removed and as the book is only 2-1/2 years old, removing what is intact and doing its job holding things together seems a bit much. BUT those two sections need to go together. Planning to cut mull to get across both sections, with enough to tuck under the lifted pastedowns on both sides. (See next posted question for view of lifted pastedowns.) Perhaps a line of glue/paste in gutter as two sections go together, and generous pasting down of the mull. Comments, please. Thanks.