The associated paper makes the assumption that for every division of rent, some room is acceptable. However, the lemma itself makes no such demand, but only guarantees the existence of an approximate rent division. The algorithm given refines the rent division iteratively, so an upper spend limit only means you might end up with rooms assigned and a portion of the rent unpaid. You could then fix the underfunded tenant's room and iterate on the others, but there's no longer a guarantee that the others wouldn't envy the bum's rent.
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It might.
The associated paper makes the assumption that for every division of rent, some room is acceptable. However, the lemma itself makes no such demand, but only guarantees the existence of an approximate rent division. The algorithm given refines the rent division iteratively, so an upper spend limit only means you might end up with rooms assigned and a portion of the rent unpaid. You could then fix the underfunded tenant's room and iterate on the others, but there's no longer a guarantee that the others wouldn't envy the bum's rent.