Rather, mangé occupies the same grammatical position as does on the table in this sentence:

Ils auraient eu bientôt sur la table tout le pauvre patrimoine.

From the sentence, we don't find out who would have eaten the patrimony. We only learn that a notary or a lawyer would have caused it to be eaten perhaps by some third party. (If we thought the notary or the lawyer would be the one to do the eating, that would be a guess, not what the sentence says.)

The following formulation would however tell us who would do the eating, namely, the notary and the lawyer themselves.

Ils auraient bientôt mangé tout le pauvre patrimoine

If possible, I would appreciate an answer that takes each of these four items and confirms or denies it.

Background

Please forgive 2 being non-sensical. If I knew how to say tied up in litigation in French, I would have said that.

Thank you. Citations to external sources are always helpful. On the original sentence, is there no alternative construction by which auraient eu is the verb form and mangé a complement? Consider: "The emperor would have had the rebels crucified." Or, "he would have had hanged and quartered whoever was responsible for the insurrection." Here we may want to say that crucified or hanged and quartered is not part of the verb. This doesn't work for the Perrault sentence even as a less favored alternative?
– CatomicMay 20 '16 at 9:34

@Catomic "is there no alternative construction by which auraient eu is the verb form and mangé a complement?": I can't see/imagine any whatsoever.
– LaureMay 20 '16 at 9:38

I certainly don't want it deleted. But back on substance, the English form have + someone/something + past participle, e.g. have the table cleared or have him prepared for surgery, does not have a French analog?
– CatomicMay 20 '16 at 9:44

@Catomic I feel you are asking a different question altogether. "have + someone/something" in French is rendered with "faire + verb in the infinitive". Faites-les entrer. Faites nettoyer la salle.., etc... Does this answer your query ?
– LaureMay 20 '16 at 9:52

The original sentence translates to "They soon would have had eaten the small inheritance" (I know small doesnt fit well here for "tout le pauvre" but let's try to keep things simple for both of us :) ).

Now your #4 translates to "They soon would have eaten the small inheritance".

What do we know in the context ?
The first sentence tells us the meunier (the miller) of the story just died and his three children must share his goods.
His legacy to his three children include his mill, a donkey and a cat. In the seconde sentence, it is explained there was no need of a notary or any lawyer : the children already divided the inheritance between themselves (les partages furent bientot faits, "soon the inheritance was divided").

And then we get to the third sentence. Here, the verbe manger is clearly not about food. It would not make sense as no food was mentioned anywhere. This is where a Native speaker may help, because this is a really informal sentence (I would say even a bit slang).

We already know the inheritance is really not consistent : a mill and two animals (of which a cat, which is useless from a work point of view). Mangé here rather means the inheritance is so small the three children divided it easily and consumed it. It gets clearer with the last sentence : the a^iné (older) took the mill, the second the donkey and the last brother, only the cat.

What an odd tense though
As of your questions, ils auraient eu mangéis a conjugated form of manger (infinitve). Mangé is the participe passé (participle past).

If you wonder about the tense used, which I reckon you do, please note this form of conjugation is called a temps sur-composé (and literally no one uses it).
It is a tense where the auxiliary is not associated with a simple tense but a composed one.
For instance, the "correct" tense one Native would use here is ils auraient mangé, which is conditionnel passé. You can find more documentation here (sorry I could not find any in English, for this is a clearly underused tense and unknown to mostly people) : https://fr.wikipedia.org/wiki/Conditionnel_pass%C3%A9_(conjugaison_fran%C3%A7aise)

In short, here are some points I wish to clarify regarding your original question :

ils auraient eu mangé is a conjugated form. The tense is called conditionnel passé surcomposé. Note no one uses this and we prefer using conditionnel passé, although it is still largely underused.

the lawyers and the notaries do not take part in the process, there is no litigation (according to the context provided; surely there can be some kind of litigation further in the book, because it is clearly an unfair division) : the children shared the inheritance between themselves without any exterior help, this is mentioned when neither the lawyer neither the notary were called (ni le notaire, ni le procureur n’y furent point appelés).

Thus, here, the sentence simply means the inheritance was so small the children did not bother having juridical help and rather shared the three items in three and gave each item to each brother regarding "importance" -- the older got the best item and the last, the worst.

Thank you. I didn't think any lawyer was going to eat the donkey; I was using "eating" to echo the French original. Please don't worry for me in that regard. As to tense, could we say that the original sentence admitted two constructions: (a) auraient eu mangé as a finite form of manger vs. (b) auraient eu as the finite form and mangé as a complement (as with sur la table). The reason I would have preferred (b) is that (a) places the (hypothetical) event of the lawyer's involvement too far back in the past, viz. in a more distant past than the time of the division of patrimony.
– CatomicMay 20 '16 at 9:22

Exactly, and this exaggeration of putting the lawyer's involvement too far back in the past (and on top of that the use of "bientot") is used by the author to emphasize how little time it would take them to eat the inheritance
– JeanBlaireMay 20 '16 at 9:31

You say, “and literally no one uses it”, other than being… obviously wrong (Charles Perrault uses it), it is also wrong to a significant extent. Some people do use it casually (sometimes abusing it) in particular in the south of France.
– Stéphane Gimenez♦May 20 '16 at 9:31

Are you also claiming that conditionel passé is not used? or did you want to include a link to surcomposition?
– Stéphane Gimenez♦May 20 '16 at 9:36

@Catomic each sentence admits more than one tense, as long as the concordance des temps is respected ! However, I think you mistake the subject here : the lawyers are not the ils in the third sentence. The ils are the three children which divided the patrimony themselves due to its smallness. The ils auraient mangé form is more appropriate in nowadays speech, because sur-composition of the tense is a bit heavy, I reckon. Although the two share the same meaning, with just a nuance of "more in the past" for the sur-composée form.
– tlombartMay 20 '16 at 9:49