Let's say the A-Hs are able to provide proof that the Black Hand provided weapons for the plot to assassinate Franz Ferdinand AND that the Black Hand was run by Serbia's intelligence office, making it too politically toxic to support Serbia in July 1914. So Serbia accepts the Ultimatum given by A-H in its entirety and thus sees the Black Hand dismantled while making Serbia a satellite of A-H, with Russian influence having been destroyed by the episode.
So WW1 doesn't kick off, Franz Ferdinand is dead, and Serbia is neutered as a threat to A-H and instead gets linked to A-H economically, while Russian influence in the Balkans is wiped out. Romania eventually would shift toward the Entente based on its desire to annex Transsylvania, but wasn't interested in joining the Entente officially; still this isn't a client that Russia can influence to the extent they did with Serbia.
A-H is stabilized by the death of Franz Ferdinand and the removal of Serbia as a threat. Instead A-H can focus her attention on the larger enemy of Russia and potentially Italy. What does this mean going forward? The A-Hs and Germans were still encouraging Polish and Ukrainian separatists, though without much success. Is there a potential flash point there? What about Russian long term stability? Can we really say that WW1 has been avoided if war is avoided over the Serbian and Franz Ferdinand issues? The Balkans is going to be locked down for the foreseeable future, so no threat there. Germany is going to have her window to start a war shut by 1916, which will scrap the Schlieffen Plan based on Russian rail construction. France and Germany had no pressing issues, even with the Alsace issue simmering in the political background (within a generation it would no longer be even a simmering issue). Britain and Germany were not longer at immediate odds over anything thanks to the Naval Race ending in 1912 when the Germans stopped passing naval bills.
In the future there could be issues over the completion of the Berlin-Baghdad rail line, the position of Christians in the Ottoman Empire, Russian political instability, A-H political issues upon Franz Josef's death, A-H issues during the 1917 Ausgleich negotiations (the Hungarians wanted defacto independence during that one, which would mean a potential Civil War or at least the occupation of Budapest), or even German issues stemming from Socialist demands for a modern constitution.
What does all this mean folks?
Providing proof would mean that the Black Hand actually WAS run by the intelligence service. Which i really, really doubt. Were there low level connexions and support? Probably. Was there high level unofficial support? Maybe. Official support? I doubt it. Contolled? No.
Otl, ah made the ultimatum impossible for the Serbians to accept, insultingly so. They didnt want an investigation, they wanted war.
If the ah demands had been merely intusive, condescending and insulting, rather than impossible, Serbia would likely have accepted them. Otl, they accepted 10 of 12 of the articles, were prepared to discuss another and wrre constitutionally prevented from acceptingthe last. Iirc.
If Serbia accepts revised demands, theres no war, and russias support is mostly diplomatic.
If Ah makes impossible demands, and goes to war, Russia pretty much HAS to join.
Best best for a limited war is romania jumps in on Serbias sise, and russia can attack/support through romania, providing defensive measures only on her border. If Russia does this, which is not necessarily going to happe, and IF she loudly proclaims that thats what shes doing, ditto, then Germany has a good excuse to back out of attacking Russia. Again, even those efforts on Russias part might not be enough, as Germany was looking for an excuse to take down Russia, rather than an excuse to get out of fighting.
Still, things might be slowed down enough that british efforts at mediation might at least be tried. And if people start realising how bloody the war could be, again, not probable, we might avoid it.
But once ah makes impossible demands and invades serbia, i dont see any way to keep russia out.