Causative scoping over disjunction

Also from Bruening 2018, citing Kuroda 2003, is this example:

花子 が 正男 に 家 を 掃除する か 部屋代 を 払わせる こと にした
Hanako ga Masao ni uti o soozisuru ka heya-dai o haraw-aseru koto ni sita.
Hanako NOM Masao DAT house ACC clean or room-rent ACC pay-CAUS that to do
`Hanako decided to make Masao clean the house or pay room rent.’

(The glosses and transliteration are per Bruening; standard orthography is my best guess.)

What’s interesting here is that the causative is analyzed as morphological (i.e. attaching only to haraw-) and yet it scopes over the disjunction. Most of Bruening’s putative “scope” problems with lexicalist analyses are non-problems in a framework that doesn’t take the syntax tree itself as a semantic representation, but this one seems stickier. Any thoughts?