You can dissolve or merge your river lines into a single line feature, then pairwise intersect of the transect lines to point FC.
Can then Split Line At Point GP tool to split the new river line where the transects cross.
Then use the Generate Transects Along Lines tool on this new FC, and check the Generate transects at line start and end, but give a Distance between transect length that is longer than the longest split river line segment.
This will make perpendicular transects at the split location. You will want to verify them, as if the split is right on a vertex where angle of line changes, not sure which segment sharing the vertex is perpendicular to.
However, this will make all the transect lines the same size (as set in the GP tool).
R_