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Túrin spoke the Mannish tongue of the House of Hador and also Sindarin (in a noticeably archaic Doriathrim way). On the other hand the petty-dwarves being largely isolationist and hating the Eldar, would be unlikely to know Sindarin, and it is also hard for me to imagine that they would speak Mannish tongues well.

Yet Túrin (and even his men) had no linguistic hardship when communicating with the petty-dwarves; in what language was this?

Is there really any other possibility, possibly Túrin, owing to the Khuzdul influences on his mother-tongue (that the Hadorians are said to have acquired before passing to Beleriand) and perhaps by conversing with Dwarven craftsmen visiting Menegroth, knew enough Khuzdul to somehow understand Mîm? Or did the petty-dwarves profess knowledge of Sindarin or some Edain pidgin?

Most likely Sindarin

It was certainly not Khuzdul, as Mîm says they do not teach that tongue, futher, Mim is described as initially speaking a foreign language:

They have no name, save in the dwarf-tongue, which we do not teach

They led the old Dwarf away to their dismal camp, and as he went he muttered in a strange tongue that seemed harsh with ancient hatred;
_{The Unfinished Tales, Part 1: The First Age, II: Narn I Hîn Húrin}

Mîm also knew the name of where is house was located, naming it in Sindarin and complaining about the changes of the names since the Elves came.

Then Mîm answered: ‘High above the lands lies the house of Mîm, upon the great hill; Amon Rûdh is that hill called now, since the Elves changed all the names.
_{The Silmarillion, Chapter 21: Of Túrin Turambar}

Mîm is also shown to have been able to notce that Túrin spoke Sindarin and was raised by the Eldar:

One thing (as Mîm observed) of which Turin never rid himself, despite his grievance against Doriath, was the speech he had acquired during his fostering. Though a Man, he spoke like an Elf of the Hidden Kingdom...
_{The History of Middle-earth, The War of the Jewels, Part Three: The Wanderings of Húrin and Other Writings not forming part of the Quenta Silmarillion: I. The Wanderings of Húrin}

Beleg was also able to call after Mîm as he fled in Sindarin:

The vengeance of the house of Hador will find you yet!
_ibid.

It would appear Mîm did indeed speak Sindarin, given this passage from Narn i Hîn Húrin from Unfinished Tales:

Then Mîm clapsed Túrin about his knees, saying: "Mîm will be your friend, lord. At first I thought you were an Elf, by your speech and your voice; but if you are a Man, that is better. Mîm does not love Elves."

This indicates Túrin was most likely speaking Sindarin at the time. Since there's no indication that he ever switched languages during the episode, I would conclude that the entire conversation takes place in Sindarin.