'Mrs. William Paul' is not synonymous with 'Mme William Paul.' According to William Paul's third edition of The Rose Garden (1872), 'Mrs. William Paul' is the Hybrid Perpetual, not the Moss. Paul lists both roses, 'Mme William Paul' under mosses, and 'Mrs William Paul' under HPs.

Simon and Cochet (1906) in their Nomenclature list both roses as well in their separate classes, but being French, they call the Mrs. a Madame, and therein possibly lies the confusion.
Paul, Simon & Cochet, and Cranston (Directions of the Rose, 1888) describe the flower of 'Mme Wm Paul' as "bright rose", which indeed mine is. Paul and Simon & Cochet respectively describe 'Mrs. Wm Paul' as "bright violet red shaded fiery red" and "red violet." (However, Sanders in 1899 who lists only the HP, describes it as "bright rose," apparently having already confused the two roses.)
Given Wm Paul's reputation, his knowledge, and his care in writing, I believe the solution lies with the master in The Rose Garden. Thus, if you accept this, you would remove 'Mrs. William Paul' from the moss entry and place it with the Hybrid Perpetual entry.