Could it be Princess Louise Pauline Sidonie of Arenberg (* 18.12.1820 Bruxelles - + 11.5.1853 Bruxelles)? She was the great granddaughter of Duke Max's great grandfather, Duke Charles Marie Raymond of Arenberg. So she was Max's second cousin, the girls' second cousin once removed, and being some years older, perhaps thought of as an aunt?
Do we know how long it was customary to porter le deuil ? I think it was one year for close relatives, but much shorter for more distant relatives and hardly at all for children, the infant mortality being so high. As far as I can see, no-one in their immediate family had died within the year August 1852 - August 1853.
But the Grand Duke of Saxe-Weimar died the 8.7.1853. As his mother was a Princess of Hesse-Darmstadt and his grandmother, a Princess of Zweibrücken-Birkenfeld line, was King Maximilian I of Bavaria's aunt, he would perhaps also be a candidate, as he was Duchess Ludovika's second cousin and the person in question once more the girls second cousin once removed.
And no doubt the whole Bavarian court would mourn a fellow, related sovereign far more seriously than an unmarried aristocratic lady realted to a side branch.