The idea behind the remark quoted in the question was that, in situations ordinarily treated with proper-class forcing (e.g., Easton's theorem), the work can be transcribed rather routinely into a Feferman-style set theory (ZFC plus a constant $\kappa$ for an ordinal and axioms saying, one formula at a time, that $V_\kappa$ is an elementary submodel of the universe $V$). Just do with $V_\kappa$ what you would otherwise have done with $V$. Where (for example) Easton got arbitrary cardinal exponentiation at all regular cardinals, you'd now get arbitrary cardinal exponentiation only at all regular cardinals below $\kappa$, but that's "morally" or "intuitively" the same (and gives the same relative consistency result) because of the elementarity of $V_\kappa$ in $V$. This "large set" approach allows you to work with the framework of Boolean-valued models rather than forcing, whereas a proper-class forcing would, in general, need super-classes (yet another level higher in the cumulative hierarchy) to do this.
In both frameworks, the real issue is not whether you work with large (i.e., $\kappa$-sized or bigger) sets or with proper classes but rather what additional conditions you impose on your forcing notions (or Boolean-valued models). As Nate pointed out, you need some conditions (in either framework) to make sure you get a model of ZFC. If you just go blindly ahead (in either forcing), you could, for example, add a proper class (respectively a $\kappa$-sized family) of Cohen reals, so that the continuum will no longer be a set in your forcing extension (of $V$, respectively $V_\kappa$). Or you might collapse all the cardinals (resp. all the cardinals below $\kappa$).
Of course, some people might want to sacrifice (part of) ZFC and work with such "strange" models. If I remember correctly, Rudy Rucker once (before he turned to science-fiction writing) proposed working in the theory obtained from ZFC by deleting the power set axiom and adding Martin's Axiom for arbitrarily large collections of dense sets (so the continuum has to be a proper class). But here again, it seems to me that it makes little difference which framework you use.
Also, I recall that Sy Friedman did some work on super-class forcing. I don't know any of the details, but I would expect that this too can be easily recast in terms of forcing over a Feferman-style $V_\kappa$.
Finally, let me mention that, if you force in the Feferman framework, you actually have two choices for what should be the generic extension of $V_\kappa$. One is to take the elements of rank below $\kappa$ in the generic extension of the full universe. The other is to take the denotations of names whose rank is below $\kappa$. The two seem to coincide in nice cases, but I don't see any reason for them to coincide in general. (The second is what corresponds to proper-class forcing over $V$.)
Although your notation $\omega_1^{CK}(\text{Ord}^M)$ suggests that you have some sort of generalized computability in mind, I'll take the question as being about all the well-orders of $\text{Ord}^M$ that are parametrically first-order definable over $M$. Such well-orders are elements of the next admissible set $M^+$, so the height of $M^+$ is an upper bound for their order-types. It might well be the least upper bound, but I'm not sure about that.
Best Answer
Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements.
The following theorem is an immediate consequence of the main theorem.
Theorem. Assume Kelley-Morse set theory with urelements KMU, with the abundant atom axiom and second-order reflection. Then there is a stationary proper class of measurable cardinals, partially supercompact cardinals, and more.
I mention the theorem because the main proof method is to undertake the unrolling construction, which produces sets at ranks higher than Ord.
Starting at lower right in the model $\langle V(A),\in,\mathcal{V}\rangle$ in which the hypothesis is satisfied, we undertake unrolling to produce the models $W$ and $\bar V$, in which there is a supercompact cardinal. In fact, it is the cardinal $\kappa=\text{Ord}^V$ that becomes supercompact in these taller models. And the supercompactness of $\kappa$ in $\bar V$ and $W$ reflects to diverse consequences in the original universe $V(A)$ and its class of pure sets $V$, such as a stationary proper class of measurable and partially supercompact cardinals, as much supercompactness as desired.