The gender of French nouns never changes as you assume they should do.

The function in the sentence is irrelevant to the gender of nouns: différence is a feminine noun, so it will always be la différence, be it subject, direct object, indirect object or whatever else.

Similarly, body parts, such as cul and coude, will also keep their own gender, no matter what is the gender of the person they belong to. So you will say le coude, le cul, la bouche, la tête, etc., whether the “owner” is male or female.

You are not yet familiar with basic syntax and you are making a surprising beginner’s error (but after all, any eventuality is as legitimate as any other if one takes into account that the errors are those of someone who doesn’t yet know): there does not exist a rule of agreement between subject and object as far as gender and number are concerned, and such a rule does not make any sense; fortunately, grammar is not like that, neither in French nor in English! Nouns keep their gender, always. Only the adjectives are subject to that rule of agreement with the subject, but not in any case; the adjective must be describing the subject, must be related to the subject through the verb in other words, as shown below.

• Elle est grande. (subject: “Elle“)
• Ce pays n’est pas grand.
• Les voiles de ce bateau sont grandes. (subject: “voiles“)
• La baguette de pain est longue. (subject: “baguette“)
• Les fleurs étaient belles mais un peu fanées et ne lui plaisaient pas ; elle les jeta.

In your sentence there is no problem of gender neutral phrasing. The problem lies in whether to make a general statement or one that applies to someone in particular. Your English formulation is not quite usual but it’s apparent that you intend a statement that applies generally; for that purpose you use habitually a possessive.

• “not to know/tell one’s ass/arse from one’s elbow”
  (dictionary definition which you then have to apply in substituting the proper pronoun for “one”)

So, speaking generally or to someone in particular you’d say

• (2) “You don’t know your arse from you elbow.” (either gender neutral (general) or not (someone in particular, but gender is not represented in the spelling); in both cases the formulation is gender neutral because there is no difference in spelling for the two genders.),

or speaking about someone you’d say

• (1) “He doesn’t know his ass/arse from his elbow.”, “She doesn’t know her ass/arse from her elbow.”, and so on for more than one unique person.

Speaking generally again, you’d say also, but not so often as “one” is not used so much nowadays,

• (2) One doesn’t know one’s ass/arse from one’s elbow. (We do have here a gender neutral pronoun.)

A specific case arises when you don’t speak generally but about one unique person and you don’t know that person’s gender. In this case “one” won’t do as it is an indefinite pronoun.

• (3) This person doesn’t know their arse from their elbow.

This exemple sentence before the last one is the one for which you want a translation. You wouldn’t translate that literally, because it is an idiom in English and in French it would not be an idiom, although it is self-explanatory and many would be those to know what you are saying. Suppose however that a context is found for the literal translation. Then you could use the same syntax as in English, that is a structure that involves a possessive.

• On ne voit pas la différence entre son cul et son coude. (“son” is both feminine and masculine, so there is no gender neutral question to debate.)

Here, your choice (“savoir”, “sait”) is not too good : ngram. “Voir” seems preferable.

However, in French there exists also the usual generic use of the definite article as option to replace a possessive and the following translation (literal), is equivalent. (You got the essentially the right idea.)

• On ne voit pas la différence entre le cul et le coude.
  (Here, you avoid altogether the gender neutral question as there is no gender involved as to whose body parts you refer to, those body parts are implied in the statment generically.)

Note : At least in British English the word “ass/are” is considered a taboo word, too coarse to be used in polite language. Similarly, in French, “cul” is very colloquial or coarse (“trivial.” in French).

(TLFi) trivial. Qui est grossier, vulgaire; qui concerne les éléments qu'une société condamne comme étant contraires aux bonnes mœurs, au bon usage, à la bienséance.

There are idiomatic translations (Harrap dictionary); they are not ideal though as their register is that of the standard language.

• (1) He doesn’t know his ass/arse from his elbow.
  Il ne sait pas où il en est. — Elle ne sait pas où elle en est.

• (2) You don’t know your arse from you elbow.” — and— One doesn’t know one’s ass/arse from one’s elbow.
  On ne sait pas où on en est.

• (3) This person doesn’t know their arse from their elbow.
  Cette personne ne sait pas où elle en est.