That’s my thought as well: rather than try to invisibly repair the cut, insert a brass accent at that point.
Option two would be to fabricate a new strap to replace the whole of the current strap.
Option three would be to unstitch about 2” from either side of the cut and shorten one end of either side such that the cut was offset between layers. Skive the leather (or at least cut at a 45 degree angle to the surface), align the overlapped ends, glue with Barge contact cement, and restitch using the existing stitch holes.
This awesome ASCII art should clarify:
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You could try option 3 first, as all it costs is time, then try option 1 if not happy with the results, and try option 2 as a last resort.
Key is to unstitch enough to allow for a strong, staggered join.