This does seem to be the case. Since the story is set in Paris, we can look at some relevant info.
A reference relates fear of bathing to the plague, spoken of here:
The habit of bathing took another big hit during the 14th century
when medical experts at the Sorbonne in Paris declared washing a
health concern. Warm water opened pores, and so could increase a
person’s risk of contracting the bubonic plague, they claimed
(incorrectly). A fear of hot water and bathing persisted for the next
Starting a little later, in the 18th century:
In the 18th century only the nobility and wealthy had bathtubs in
their homes, at the Marais and Faubourg Saint-Germain, the fashionable
districts of the time. Other Parisians either did not bathe at all,
bathed with a bucket, or went to one of the public bath houses, which
provided hot tubs of water for a fee. They were heavily taxed by the
government, and only a dozen survived until the end of the century.
This article seems to infer that this attitude continued, at least into the beginnings of the 19th century:
Only at the beginning of the 19th century did the idea of taking a
regular bath as a part of personal hygiene begin to take shape. It
made a slow progress in the upper classes, but the common people
remained blissfully dirty.
(all emphasis mine)
So we can see that bathing in general was uncommon, and more prevalent among the wealthy than the poor, agreeing the the questioners premise. It is worth noting that the association of wealth or privilege with access to bathing is not unique to this time, but dates back even to the Roman period, and was previously discussed here: