P. Dehornoy, M. Dyer and C. Hohlweg have shown that every finitely generated Artin-Tits monoid admits a finite Garside family. We investigate the same problem for a generalized class of Artin-Tits monoids introduced by D. Krammer. Our focus is on a distinguished family of monoids that are not Garside monoids yet admit finite Garside families right-bounded by a distinguished element $\Delta$.