Which others ? There is at most one local system on a unipotent class for which the center acts by \chi.

Yes that's right, but he says that for all unipotent classes there is at most one in \mathcal{M}_{\chi}. So, the one in the class parametrized by (3,7,...) or (1,5,...) is cuspidal. Isn't it ?

In Lusztig's article, before Proposition 14.4. he explains that these local systems correspond to spinorial representations referring to Bourbaki. So, is not that enough ?

Marko Tadic also did that for tempered representations in : On tempered and square integrable representations of classical p-adic groups. (English summary) Sci. China Math. 56 (2013), no. 11, 2273–2313.

Colette Moeglin, Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc. (JEMS) 4 (2002), no. 2, 143–200. Colette Moeglin, Marko Tadić, Construction of discrete series for classical p-adic groups. J. Amer. Math. Soc. 15 (2002), no. 3, 715–786. Chris Jantzen, Tempered representations for classical p-adic groups. (English summary) Manuscripta Math. 145 (2014), no. 3-4, 319–387.