If you can't agree, you escalate to a higher, uninvolved authority, up to the High King if necessary (if you disagree with a previous authority, it's not uninvolved anymore, though if it's for bullshit reasons people won't be amused). And no, the onus isn't on the accused to prove innocent. The dwarf does have to prove their accusation. Humans would get less trust in this, but probably would also be much more willing to fuck around with the process, and as things escalate and higher authorities get involved, you're going to find someone who isn't going to lie, because the dwarf alliance is simply too important. Even if that means the Emperor has to kick some Elector Counts ass.
As seen in the quote I provided in the post of mine you're quoting, if the result of discussion between the Elders of the Clans is that one Dwarf is in the wrong they will accept that result and not escalate the situation further even if they may grumble quietly about it.
Not every problem can be kicked up the chain infinitely, when it doesn't contain matters worth the effort and attention
Bokri the cobbler isn't going to present his case over whether he was shortchanged by a shilling by a couple travellers in Middenheim who left the city that same day and are now who knows where, to the whole of the Karaz Ankor, they have much bigger concerns to occupy their available time with
He's not even going to present his case before the local town guard, because what are they supposed to do about it? Go haring off into the countryside shaking down every pair of travelers they see?
Nor for the matter would having the matter escalated to a higher court do much to actually determine guilt in the matter, in this case
Because this is entirely in the realm of "he said, she said"
Did he get shortchanged? Was it deliberate? Did they actually hand over the correct amount of money but then it got swiped by someone else who nobody noticed somewhere down the line? Did he accidentally drop an extra coin when paying his tab and come to the wrong conclusion when he was counting his ledgers the next morning? Was he shortchanged by
someone else that same day but assumes the missing money must be because of those two because they just seemed shifty to him?
This case isn't getting solved, it's too inconsequential, the details are too far in the past by the time attention will be drawn to it, and that money might as well have vanished into thin air
This is just going to go into Bokri's personal shitlist, and if he ever sees those two bastards again (or possibly two people who have the misfortune of being mistaken for them) he'll be having words with them on his own
It just goes straight into The Book
Not The Big Book, probably not even The Moderately Sized Book
His own personal Small Book
Someone is still wrong in the event a mistaken accusation, that person is either a third party who deceived them, or it's the Dwarf themselves
A very considerable amount of Dwarven pride means that they are very loathe to believe that it's themselves who were in the wrong, and the more they entrench the greater the shame if it were to be proven that they are mistaken
Even in the case of accepting your Elders working out a way to bury things on your behalf (something that is not always going to be an option), in your mind the other guy is still in the wrong
You're just a noble martyr for bearing this burden for the good of your clan