Modalities in general do not commute with negation. So for (a clear) example, "agent doesn't know that X [is the case]" and "agent knows that not X [is the case]" are obviously different so we would not think to propose that K~x and ~Kx are the same/equivalent.
As a TLDR summary of equivalences for the (relevant part of the) deontic logic vs English:
You must not leave [the spoon at the bottom of the container].
It is mandatory that you do not leave [the spoon at the bottom of the container].
You need not leave [the spoon at the bottom of the container].
You don't have to leave [the spoon at the bottom of the container]
It is not mandatory that you leave [the spoon at the bottom of the container].
As well known to linguists must and have to (in English) accept positive (non-negated) statements under their scope as well, but "need" (not be confused with "need[s] to") does not, i.e. "John need leave" is ungrammatical but "John must leave" is ok.
The issue with imperatives (unlike modals more generally) is that we don't seem to have much meaning attached to their negation, in contrast to what's negated inside them.
Since a precise (linguistic) mode of "Do not leave ... due to forgetfulness" is apparently not central to the question, we can conceive it as "do not forget the spoon at the bottom of the container" (per Conifold's answer). Furthermore, you can simply think of "do not forget" as "remember to" then.
- remember to breathe: Rb.
- remember to add salt to the dish: RA(s,d).
- remember not to overcook the meat: R~O(m).
But what does it mean to negate an imperative? Surely we can formulate these in a logic just as easily as we can in the "know" (epistemic logic) case, but with the negation of imperatives we seem to have trouble interpeting them; they sound non-sensical as imperatives in natural language, except maybe as pun admonitions:
- ~Rb: "don't remember to breathe" or "forget to breathe".
- ~RA(s,d): "don't remember to add salt..." or "forget to add salt..."
- ~R~O(m): "don't remember not to overcook the meat" or "forget not to overcook the meat)
Note that this is not the same linguistincally as the "forget about" idiom... althout maybe we could read:
- ~Rb: "forget about breathing"
- ~RA(s,d): "forget about adding salt to the dish"
- ~R~O(m): "forget about not overcooking the meat"
These did require some grammatical tweaks though. But the more troublesome part is that the longer sentences, which use predicates under the modality (which linguistically correspond to transitive verbs), already seem to have somewhat unclear meaning, even in this interpretation. Or at least they sound awkward to me. So, this is why I was hesitant about interpreting "don't leave ... due to forgetfulness" just as "forget about ...".
Likewise one can interpret "don't forget" deontically as "impermissible (forbidden, prohibited)".
If we take the "ought to be the case" / "it is obligatory that" as the basic deontic modal primitive (aka deontic necessity denoted by O) as it's commonly done in "standard" deontic logic, then impermissible is defined as Ix = O~x. (And permissible as Px = ~O~x = ~Ix.)
However ~Ox is read as "it is omissible that". Which is fairly award and seldom heard in natural language. (Basically it's the same semantic weirdness as [externally] negating imperatives.) SEP (consequently) also notes that
Deontic non-necessity [...] is seldom labeled [as separate operator].
It's more natural to hear "it is optional that", but that one is actually is (~Ox /\ Px), i.e. "omissible and permissible". In a nutshell:
Frankly, I have serious doubts that if you pick untrained people they would make that kind of semantic distinction between "optional" and "omissible" as mandated in deontic logic. (Although I've done a bit of searching on this, there don't seem be any empirical studies on this particular issue, unlike many other in the logic-psychology interface.)
Anyway, since you mentioned first-order logic... there is actually something to be said here about the general relationship between modal operators as an analogy to models of quantification in [many sorted] first-order logic on non-empty domains (i.e. Aristotelian existential import). Best illustrated in the diagram below:
As noted in SEP this observation basically gives the Kripke semantics of modal logics (in general). Formulated in terms of deontic aspects, using serial frames (i.e. if something is obligatory then it is permissible too), that's
Thus, p is obligatory iff it holds in all the i-acceptable worlds, permissible iff it holds in some such world, impermissible iff it holds in no such world, omissible iff its negation holds in some such world, optional iff
p holds in some such world, and so does ¬p, and non-optional when p
either holds at all such worlds or at none. If a formula is true at every world in any such model of serially-related worlds, then the formula is valid.
Interestingly enough, in Chinese it seems one has almost the exact formulation of this distinction from a modal logic:
Exemption vs. Prohibition (¬□/□¬)
Generally speaking, the insertion of negation confers a specific pragmatic meaning to a proposition, in some cases producing a shift from the propositional to the illocutionary level, thus generating a speech act. More specifically – as observed in deontic logic (Von Wright 1963: 136ff) and in Chinese linguistics investigation (Li Jinxi 1924) – once a normative statement is turned to the negative form, it produces two antithetic and irreducible sentences, either Prohibition (‘it is necessary not to’) or Exemption (‘it is not necessary to’), either bùkě (‘it is not allowed [possible] to’) or kěbù (‘it is allowed [possible] not to’) (Li 1998–1924: 104–105).
The rest of the paper is much less clear to me (as both as non-linguist and not a speaker of Chinese)... but it seems to argue that some Chinese expressions don't actually change their meaning by this kind of swapping, i.e. they have a preferential reading... and there are "suppletion strategies" to actually give them a different reading (i.e. one that would reflect true swapping of the negation position at logical level.) So it looks like all/most natural languages are messy enough that we can't always find the neat reflection of (deontic) logic we might hope for.
The (roughly) equivalent situation in English seems to be that some modals lexically swap position with negation:
John does not have to leave.
It is not mandatory for John to leave.
I.e. the lexical order is the same as the logical/semantic order.
John may not leave.
It is not permissible for John to leave.
Here the lexical order in "may not" is actually the opposite of the logical/semantic order (in the longer sentence with the same meaning).
Also, if you search the web for "it is omissible that" in quotes it seems you find nothing but (deontic) logic pages. The more natural English language expressions seem to be "it is not mandatory to" or "it is not required to".