English equivalent for French "assaut de courtoisie"?

Question

On StackExchange, after we've asked a question and got an answer, or ourselves posted an answer, it frequently happens that a complementary discussion is established through comments, where we exchange precisions.
This commonly stops after a few comments have been posted from each one.

But in some occasions one comment includes kind words, really beyond a simple "Thanks", so per common courtesy habit the recipient has the feeling he can't remain without sending kind words in turn.
And sometimes this feedback itself causes the previous sender to feel the same... then we may fall in an awkward situation where it becomes as difficult to stop (because of the courtesy reflex) as to continue (because it might never stop, and anyway it's strongly discouraged by the StackExchange policy).
We can also note that this situation is more likely to happen in sections such as English language learners and other "soft skills" related ones, while for instance StackOverflow or CodeReview are rarely involved.

Just now, I felt into such a situation (in the French language section), and planned to gently express that we both had to avoid persisting so.
And that's when I realized that I didn't know how to translate the French expression that was me naturally came to mind: "Nous devons cesser de faire assaut de courtoisie".

In French this "assaut de courtoisie" is a well established formula, but I strongly doubt it could be translated literally (even if Google, imperturbable, quietly proposes "assault courtesy", I don't trust it!)

EDIT (because I realized I didn't clarify the exact meaning of the formula in French, so it may be misinterpreted): here "assaut" doesn't carry an hostile sense like when, for instance, an army launches an attack.
It rather refers to the fencing domain, where the agressive aspect is only due to competitive spirit.
In addition, the "assaut de courtoisie" is not one-way: the expression implicitly means that each is launching his assault in turn (again like in fencng domain).

Answer 1

(please hang on until the edited end of this ridiculously long answer to see my preferred option [which I've changed in response to your edit]!)

Such behavior reminds me a little of Warner Brothers’ Mac and/et Tosh, the two overly-courteous “Goofy Gophers” (often confused with Disney’s chipmunks “Chip-n-Dale/[Tic et Tac in French]) whose

... ridiculous … over-politeness [often] prevent[ed] their ability to get on with the task at hand.

(from Wikipedia)

In my opinion, whenever “over/ly” (or “too/too much”) is added to an otherwise positive trait or behavior (as above with “polite/ness”), the positive notion tends to become seen as a negative one (as in “too much of a good thing”).
To the extent that adding “assaut de” might do the same thing to otherwise positive traits (such as “courtoisie”) in French, you could capture this negativity in the specific context of politeness/courtesy by going full-blown negative with the single word “sycophant/sycophancy/sycophantic” , but in the context of “StackExchange” comments/exchanges, this word would really only make sense when the “over-politeness” is directed to a “Moderator.”

SYCOPHANT [count] formal + disapproving
: a person who praises powerful people in order to get their approval
Sycophancy noun [noncount]
"Her praise was obvious sycophancy."
Sycophantic adjective [more sycophantic; most sycophantic]
sycophantic praise/flattery

(from Merriam-Webster – Learner’s Dictionary)

For a verbal phrase that maintains (but tempers a bit, I think) the negativity (and which would be more suitable for use among/between equal peers), you could consider:
“pour{ing} it on {a little/bit} thick”,
where the “it” in this case would be “the courtesy/politeness/praise” and from which you could possibly get “a thick outpouring” to use with “of praise/courtesy/politeness” as a noun phrase similar to its use with “of fear” in the linked example from Google Books.

pour it on thick
To exaggerate, aggrandize, or overstate some emotional experience, response, or appeal, such as blame, praise, flattery (emphasis added), excuses, etc.
"Jim carries on as though flattering the boss will get him a promotion, so he's always pouring it on thick for her."
"OK, Bob, I think Mary understands the trouble she's in, no need to pour it on so thick."

A more neutral (i.e., slightly less negative) way to capture the notion of a/an “[overly] thick outpouring” (and which has the added advantage of seemingly being closer to both a possible literal translation and (as I previously understood it) possible metaphorical meaning of “assaut de courtoisie”) could perhaps be found in the verbal notion of
“bombarding {somebody} with {something}”,
from which you could get a noun phrase to use in your translation, such as “a/this/our/your bombardment of praise/courtesy/kind words/admirationetc”, for example(s):

"We should [really] stop this bombardment of [mutual] praise/courtesy/admiration!"
or keeping its use as a verbal notion:
"We should [really] stop bombarding each other with [mutual] praise/admiration/etc [like this]!"

bombard somebody with something
to continually send someone something, esp. to inform or influence them
"Every day it seems as if we are bombarded with e-mail messages warning of computer viruses."
"Stuart bombarded her with flowers, phone calls, and faxes just to get her to say she would go out for dinner with him."

(from Cambridge Dictionary of American Idioms viaThe Free Dictionary by Farlex)

(example of “bombardment of praise” from Child Psychology by Dr. Anupriya Chadha, via Google Books)

Finally (and here comes my preferred option/s!), with your clarifying edit and comments in mind, I’ve given some more thought to and done some more research on the matter and I now think:
1) that the notion of a positive (i.e., good-natured/kindly) mutual exchange is contained in “assaut de” as it is used in “assaut de courtoisie” in French;
2) that this notion could be captured well in English with duel of or contest of; and
3) that either of the above could easily (and idiomatically, although I wouldn’t call them fixed idioms/expressions) be used with any of the following:
politeness (from the National Library of New Zealand);
kindness (from the Edinburgh Review); or even
courtesy itself (from CalPoly, paragraph 5) ...
... to capture somewhat both the literal and (as I now understand it) figurative meanings of “assaut de courtoisie”:

"[Maybe] we should stop/end this/our duel/contest of politeness/kindness/courtesy!"

(cf: this Reverso entry where “assaut de politesse” is translated as “rivaling in politeness,” but I humbly disagree with Reverso’s use of the gerund/present participle form of rival, and even in its noun form (rivalry), I don’t think “rivalry of politeness/kindness/courtesy” would be as idiomatic as either “duel of …” or contest of … .“