Can an exception clause be non essential?

First, let's look at the difference between an essential and nonessential:

"Essential clause: a dependent clause that is necessary to the basic meaning of the completed sentence.
➞who are pregnant
Women who are pregnant can crave salty or sweet foods.
Nonessential clause: a dependent clause that is not necessary to the basic meaning of the completed
sentence.
➞who growls whenever the phone rings
Elmo, who growls whenever the phone rings, tried to attack the vacuum cleaner."

Take, for example, this essential exception clause from the Bible in Nehemiah 4:
23 So neither I, nor my brethren, nor my servants, nor the men of the guard which followed me, none of us put off our clothes, saving that every one put them off for washing.

This is speaking of men on a work project who were under constant threat of being attacked. So not taking their clothes off at all would be the meaning if the exception clause were to be omitted. Leaving the exception clause out changes the meaning, therefore the exception clause is essential.

I am having a discussion with someone who says that ALL exception clauses, no matter what, ALWAYS have to be essential. I am looking for confirmation that that claim is completely false. It sounds as though he was taught in school that exception clauses, no matter what, in whatever context or configuration ALWAYS HAVE TO BE essential. So when I produce an exception clause that is non essential, he simply resorts back to saying all exception clauses have to be essential.

To now demonstrate a NONessential exception clause, I can take the above verse, Neh 4:23, and create an imaginary context by simply adding a sentence in front. Notice how under this entirely different context, the identical exception clause in the identical sentence is then NON essential. So the same exception clause is able to be changed from essential to nonessential by simply establishing an entirely different context:

We were instructed, by the supervisor of the work team, that the former requirement that workers undress for de-lousing at the end of each shift had been lifted. So neither I, nor my brethren, nor my servants, nor the men of the guard which followed me, none of us put off our clothes, saving that every one put them off for washing.

This contexts identifies that the putting off of their clothes that the sentence is primarily referring to is the putting them off for delousing, which makes the information that they put them off for washing, excludable, being merely an added side point.

Notice how the exclusion of the exception clause manifests that it was not "necessary to the basic meaning of the completed sentence":

We were instructed, by the supervisor of the work team, that the former requirement that workers undress for de-lousing at the end of each shift had been lifted. So neither I, nor my brethren, nor my servants, nor the men of the guard which followed me, none of us put off our clothes.

Is this not a valid example of an exception clause that is nonessential?
Please someone give me some feedback.

This is an independent clause.

Are you saying the last clause is NOT an exception clause?

The previous question by me, was not stated properly. Let me try again:
Isn't it true that in certain sentence constructions, which have a non essential exception clause; it is impossible to replace the non essential exception clause with an essential exception clause? Is it not true that in such constructions, an 'essential' exception clause cannot competently fit, no matter how imaginative the author? So that the person attempting to replace the non essential with an essential, he discovers that it is impossible: a logical fallacy is the result in every single case where such an attempt is made.

For example, I have below, a long sentence which is fashioned after a Biblical sentence from Matt 5:31,32. So the following is a rough parallel of that long sentence from Matt 5:31,32. But first some background:
Some students are mistakenly assuming that moving violations issued by campus police will not be recorded as what will effect insurance rates. The same campus police are jaywalking hawks as well. So here is the sentence with the non essential exception clause:
You have heard the rumor that getting tickets is of no concern with regard to your auto insurance cost: but I say to you, that whoever gets tickets, saving for jaywalking, causes their rates to go up; and whoever uses your vehicle and gets a ticket will make the rates to go up.

Since the audience is familiar with that OTHER, different kind of ticket, for jaywalking, which is commonly issued, which does NOT cause insurance rates to go up; the exception clause fits perfectly. The sentence starts off talking about tickets that WILL cause the rates to go up, while the exception functions as an aside, referring to a different kind of ticket that does NOT cause the rates to go up.

"Saving for jaywalking" is the non essential exception clause. Notice how when omitted altogether, the main thrust of the sentence is still in tact. Omitting that bit of added information does not take away from the main important points being made:
You have heard the rumor that getting tickets is of no concern with regard to your auto insurance cost: but I say to you, that whoever gets tickets, causes their rates to go up; and whoever uses your vehicle and gets a ticket will make the rates to go up.

Now back to the question of this paper: Can any possible "essential" exception clause replace the non essential and the sentence still make sense?
I am saying no, it is impossible. The sentence format forbids that any other than a non essential exception clause can function therein.

Try it yourself. I broke down the sentence into its 6 parts to be able to be analyzed easier. See if you can imagine any essential exception clause that can replace the non essential one, and still have the sentence function competently.
A) You have heard the rumor that getting tickets is of no concern with regard to your auto insurance cost:
B) but I say to you,
1) that whoever gets tickets,
2) saving, unless, except _________ ,
3) causes their rates to go up;
4) and whoever uses your vehicle and gets a ticket will make the rates to go up.

In every case where an attempt is made to insert an essential exception clause for line 2, chaos ensues. The result is a logical fallacy. For example:
A) You have heard the rumor that getting tickets is of no concern with regard to your auto insurance cost:
B) but I say to you,
1) that whoever gets tickets,
2) unless it is for speeding,
3) causes their rates to go up;
4) and whoever uses your vehicle and gets a ticket will make the rates to go up.

The reason no one can answer your question is that you are putting the grammar before the horse.

The horse is the meaning, and these grammatical names don't describe what we necessarily need or want in any given sentence, they simply are names that grammarians have contrived to enable them to discuss and describe how language works.

Try making your examples simpler sentences and try giving accurate grammatical descriptions and names to that which you are describing.