When I needed something similar I just picked a planet name (can't remember if I made it up or got it from a list somewhere): Koltis.

The situation on Koltis was an ongoing low intensity planetary civil war (basically the half dozen or so hive cities had split into factions and were fighting over who would control the planet, but since none of the three factions wanted to destroy what they were vying to control, it wasn't full scale field conflict, more cold war killing each other's troops far from home, using proxies and client states to hide agendas, etc. type stuff). The arbites got involved only when someone got the clever idea to try to leverage power/advantage by messing with the other side's tithe production.

The big bad in my scenario was a genestealer cult that had broadly infiltrated and had members in all factions in the war, and had basically orchestrated the whole thing on all sides so that they could remove the extant planetary governor and all the important folks from the various on-world power players who might oppose their secret takeover and be able to blame it all on unrest and civil war.

The manipulator's hoped the Arbites would remove and execute the Governor for incompetence for letting the tithes get affected by the conflict., and their puppet would be positioned to step in.

Game never got quite that far b/c that entire acolyte team committed a cascade stupidity resulting in serial fate burn. (i.e. they'd burn fate to survive one debacle, then run out and do something newly suicidal) No one was left alive to uncover the conspiracy.

If you want to have Adeptus Arbites in great numbers but avoid Guard why not set up a situation where a notable Imperial Hero has been given Govenorship of a world for a lifetime of good service. A few years down the line said Govenor rebels. The Imperium send in a Guard regiment who promptly join forces with him. The Adeptus Arbites realise that if control is to be regained then they are going to have to step into the breach,

The Departmentum Munitorium is concerned that if they send another Guard Regiment they too might go over to the side of the former war hero. While the Masters of the Administratum debate over how to proceed the Arbites are forced to fight a war they aren't really equiped to fight.

At the same time the Inquisition has an ideal reason to take a very active interest in the war. Why has a well rewarded Imperial hero rebelled. Did the first Guard regiment turn traitor out of some misplaced loyalty or is there something more sinister at work? Indeed why are reports filtering through that even the Regimental Commissars seem to be siding with the rebels?

Visitor Q said:

This is a wickedly good idea and I'm planning to introduct it into the campaign one way or another. However, so far my plan is that Imperial Guard won't be involved (at least in the beginning) because the war is against lightly armed rebels and escaped mutants and rogue psykers... Adeptus Arbites isn't exactly out of their depth here, on the contrary, this sort of "street-war" is exactly the sort of thing they are equipped, trained and expected to fight. Imperial Guard is what you call in when Arbiter standard issue carapace armor, shotguns and rhinos with that nice, fluffy stormbolter isn't enough.

I'll elaborate this with a 40k refit of the old "snake model" joke:

Upon encountering a heretic on a Imperial World:

Adeptus Arbites: Drives over heretic, grins dryly, and looks for more heretics.

PDF: Makes contact with heretic, ignores all Imperial directives and Rules of Engagement by building rapport with heretic and winning its heart and mind. In the end, decides to join the heretics.

Imperial Guard: Kills heretic with massive Time On Target barrage from several hundred lasguns, three Manticores and a Baneblade. Kills several hundred adjacent citizens as unavoidable collateral damage. Mission is considered a success and all participants are awarded.

Inquisitor: Plays with heretic, then executes it.

Temple Assassin: Prepares a perfectly orchestrated infiltration operation to kill the heretic, but can't receive authorization from High Lords of Terra to engage the target.

Space Marines: Lands their drop pod on and kills the heretic.