Yes, that is a reasonable way to use the LASSO method. See Example 3 in this note which shows use of HPGENSELECT to perform LASSO selection and discusses shrinkage methods more generally. See also the Gunes paper that is linked to. It discusses LASSO and related methods further, though not in the context of categorical response models. And yes, you can use the LASSO method with a nominal, multinomial model with DIST=MULT and LINK=GLOGIT.
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