I think that the condition $(\Lambda+\rho)(\alpha) \ne 0$, $\forall\, \alpha \in \bar{\Delta}_1^{+}$ you are refering to, is the exact translation of the verbal description of (a)typicality provided in the first paragraph of the OP.
You can find a fairly detailed description of these results on Dictionary on Lie superalgebras, by Frappat, Sorba and Sciarrino, section 39, p. 58-59. (see also the two papers by Kac referenced therein; they provide more details and some proofs).
Furthermore, the character formula for the singly atypical representations of the LS $A(m,n)$ and $C(n+1)$, is provided in section 7, p. 11-12 of the same reference (see also the references included for proofs).