I know a strong accounting clause when dealing with labels is smart.
Not to put too fine a point on it, but it's not really the accounting clause you mean. That clause is very administrative in nature (e.g., how often, who/what receives the money, etc.).
I think you're referring to an audit clause which is the mechanism that allows artists, songwriters, etc. to look under the hood (where they will almost always find corrosion, albeit less so today than in the past).
Unfortunately, the legal strength of any audit provision is mitigated by practical realities.
Let's start with this one: It is very, very common that artist and songwriters tend to be non-confrontational and they view sending the audit notice to the label/publisher as something that will make them angry and disrupt the relationship (regardless of how many times I tell them it won't). So, regardless of the audit language, there is often this initial hurdle to get over.
And assuming you can get the client to move forward, it's best to avoid the following: Audits usually cost between $25,000-$60,000. YMMV, but if my client's account is unrecouped and the audit reveals a $100,000 discrepancy, but the discrepancy isn't enough to put the account into a recouped position and the client had to pay $50,000 to find out they're still unrecouped, it's usually not a victory for something you had to convince them to do in the first place.
Now let's take the audit situation for Bieber, Katy Perry, Kanye, Luke Bryan, Bruno, or whichever superstar you want to pick. It will absolutely be worth doing an audit and there's no doubt the auditor will find significant "discrepancies". However, those discrepancies are based on contractual interpretations and, as we often see with the Supreme Court, 5 judges can interpret the facts and the law one way, and 4 judges can interpret them in another. Point is: Artist says "You owe me $1M". Label says "We think it's $300,000". And from there the parties work to settle on some number they can agree on. The contractual language is merely a debatable starting point and not some ironclad legal position (because if it was, the parties wouldn't be arguing about it).
All in all, I think most accounting and audit provisions can look pretty similar if moderately competent people are involved in negotiating them, but it's the practical realities that will really control the outcome.
Is there such a thing when dealing with PROs?
No, for the very reasons that have been discussed in this thread. Unlike a record deal or publishing deal that do set forth some royalty calculations across a broad spectrum of exploitations of masters and compositions, the PROs don't provide that kind of definition and if ASCAP did provide the audit language, then everyone will argue about what they got since, as I said, it's not an exact science.
It almost sounds like you trust them implicitly
Nope, but I understand the nature and scope of what they are trying to do and believe they do a decent job inside the paradigm they are forced to operate in. Room for improvement? I've already said there is. Have they improved? Yep.
Are they arranged as for-profit or non-profit entities? (Not that that really makes them automatically more or less scrupulous either way.)
From the SESAC Wikipedia page:
"Whereas ASCAP and BMI operate on a not-for-profit basis, SESAC & Pro Music Rights[3] retains some income as profit.[4] While ASCAP, BMI, & Pro Music Rights[5] distribute all income from performance royalties to their composer and publisher affiliates (less an administrative fee), SESAC retains an undisclosed amount of performance royalty income.[citation needed] SESAC is also unique among the U.S. performing rights organizations in that it does not offer open membership – one must be approved to join.[6]"
And I think the "for profit" aspect does make a difference (although not necessarily as to being scrupulous). If you are running a business for profit, then it seems like a very conflicted situation to me. Whatever the amount of money that SESAC is keeping as profit, it would seem to reduce songwriter royalties in some way.
In theory, BMI and ASCAP take what they need to run their business on a non-profit basis and distribute the rest. Anytime you have $1B involved, there's going to be problems, but overall I think ASCAP and BMI do a decent job of keeping their overhead within reason and pursue their mission of trying to help songwriters.