Suppose $A$ and $B$ are full-rank and well-conditioned.
Is Lipschitz continuity held for generalized inverse?
$$\|A^+ - B^+\| \le \omega \|A-B\|,$$
for some $\omega > 0$, where the norm could be Frobenius norm or L2 induced norm.
In particular, if $B=PA$, where $P=VV^+$ is a projection matrix onto subspace $span(V)$. Will the following argument be held?
$$\|A A^+ - B B^+\| \le \omega \|A^+\| \cdot \|A-B\|$$
$$\|A^+ - B^+\| \le \omega \|A^+\|^2 \cdot \|A-B\|$$
for some $\omega >0$.
Locally, yes. The entries of $A^+$ are rational functions in the entries of $A$, and you assume that the denominator (e.g., $\det (A^*A) $) is bounded away from zero.
If you need a more precise or more global statement, you need to be more precise about assumptions. E.g., the family of scalar matrices $sI$, $s>0$, fits your description (condition number is $1$), but the inverse map is not globally Lipschitz, $(sI)^{-1}=s^{-1}I$.