I would like to have something like a linear order on classes, such that every instantiated predicate of classes has a minimal instance in that order. For my purposes, it is fine to assume V=L for sets.
In the context of models of second-order arithmetic, Mostowski showed that starting with the theory $Z_2$ (the second-order arithmetic analogue of Kelley-Morse) together with the scheme of dependent choices, one can use forcing to add a well-ordering relation on classes so that the full second-order comprehension scheme continues to hold in the language with the new relation. The argument appeared in his paper "Models of second order arithmetic with definable Skolem functions" [Fund. Math. 75 (1972), 223–234] and was later corrected in "Erratum to the paper `Models of second order arithmetic with definable Skolem functions'" [Fund. Math. 84 (1974), 173]. Initially, Mostowski used only the choice scheme to prove the result.
Mostowski's argument generalizes to the theory Kelley-Morse together with the scheme of dependent choices. The scheme of dependent choices states that if a second-order definable relation has no terminal nodes, then it has an $\omega$-chain. The forcing is supposed to be akin to the forcing argument that a global well-ordering class can be added to a model of ZFC, but in this case the condition are classes, and so we have a hyperclass forcing (in Sy Friedman's terminology). I only know about this construction from conversations with Ali Enayat, who will probably give many more details when he is around.