With all due respect
The OP's current 11sp cassette is on an 11sp freehub: it would not fit on an 8/9/10sp freehub.
And a short cage mech will not 'manage' a 50-34 and 11-32 and certainly not a 11-34.

to
@MJAustin
The OP does not say what rear derailleur they have but I shall guess:
https://www.specshift.bike/components/rear-derailleurs/shimano-ultegra-rd-6800-gs
The spec allows a 32t large sprocket max (the SS spec is 28t max). If the RD is 105, specs are the same.
If this rider is struggling with a 29" gear then I doubt a 27" gear (34/34) will offer a sufficient margin (if clipped in). If not clipped in they'll have a 24" gear easily available, with both feet on the road.
The solution, more expensive than a cassette, is a sub-compact chainset eg 46-30, but it'll get you up the hill.