This comes from an UN-professional perspective - but what I'm thinking is you need to spread the load over a large unsupported area.
Maybe GLUE a much larger piece of hardwood diagonally to the stairwell wall and once thoroughly dried attach the hand-rail to it (do be aware of the screw lengths and don't greatly exceed the hardwood panel). Then paint (wall-paper or stain) this piece of hardwood to blend in.
Then obviously be slow, deliberate, and careful.
YMMV.
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former 2008 Jayco Eagle 29.5 RLS
former 2014 F250 6.7 4X4, CC LB
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