I second the answers that place herbivores in the role of heavy cavalry (or, I would hazard, animal 'pikemen' - big masses of bodies that a few lions or dogs cannot really approach), or as runners, scouts etc.
But consider the social dynamics. Most carnivores are lone wolves (e.g. prey birds) or hunt in small packs. Two small packs of lions in the same territory will fight each other before they turn their attention to prey. Completely un-managable.
The strategists and generals of the army may well be intelligent herbivores, since they are used to dealing in large numbers and the movement of any army worthy of the name is necessarily the movement of a herd.
Managing an army is a man-management and foraging task. More soldiers die of disease and malnutrition than enemy action. Persuasive oratory by the general is constantly needed to prevent mutiny when the promised wages inevitably fail to materialise.
Your supreme commander is probably a Pig. He 'gets' the psychology of both plant eaters and meat eaters. He can make speeches that appeal to both.
His Aides de Camp include hares, horses, rats etc as well as carnivores. The army's chief quartermaster is a squirrel (for obvious reasons). The rats are his military police and political commissars (supported by dog packs as enforcers), they are everywhere and constantly on the lookout for sedition and poor morale.
The main body of the army - deployed in the centre - consists of a battle hardened Bovine core. They don't do much, but serve to anchor the centre and while they retain cohesion, are unlikely to be routed by a few mangy carnivores, especially since the former outnumber the latter. The Bovine blocks are probably officered by dogs to move them about effectively. This main core does not charge the enemy until their foes are on the brink of collapse, because once they let rip you ain't getting them back in formation.
The carnivores are deployed on the wings like traditional cavalry - to try and encircle the enemy centre, or as skirmishers ranged along the front to worry the main body of the enemy force, and possibly a few key shock troops kept in reserve - however this small cadre is unlikely to be decisive.
Another possible formation echoes renaissance pike and shot armies. Big blocks of dumb heavy Bovines (pikes) surrounded and supported by faster moving and more long-ranged carnivores (musketeers). When superior enemy numbers threaten, the carnivore sleeves of the animal tercio shelter under the legs of their more bulky comrades.
Substitute the pikemen for cows and the blocks of shot at the corners for carnivores. Then go read thos fabulous account of the Battle of Ceresole.
The pike and shot infantry had by this time adopted a system in which
arquebusiers and pikemen were intermingled in combined units; both the
French and the Imperial infantry contained men with firearms
interspersed in the larger columns of pikemen. This combination of
pikes and small arms made close-quarters fighting extremely
bloody. The mixed infantry was normally placed in separate
clusters, with the arquebusiers on the flanks of a central column of
pikemen; at Ceresole, however, the French infantry had been arranged
with the first rank of pikemen followed immediately by a rank of
arquebusiers, who were ordered to hold their fire until the two
columns met. Montluc, who claimed to have devised the scheme,
In this way we should kill all their captains in the front rank. But
we found that they were as ingenious as ourselves, for behind their
first line of pikes they had put pistoleers. Neither side fired till
we were touching—and then there was a wholesale slaughter: every shot
told: the whole front rank on each side went down.
Again, substitute the big bovines for pikes and the big cats for those armed with firearms.
Another bit of Ceresole from the link above that I like, which I have edited to substitute animals for men:
On the first charge, Enghien's wolfpack penetrated a corner of the
Imperial Bull-square, pushing through to the rear and losing some of
the volunteers from the Black Forrest. As bulls ranks closed
again, the wolfpack turned and made a second charge under constant
arial attack from the hawks circling above the Imperial formations.
This was far more costly, and again failed to break the Imperial
Bulls. Enghien, now joined by Dampierre's Foxes, made a third
charge, which again failed to achieve a decisive result; fewer than a
hundred of the wolves remained afterwards.
Enghien believed the battle to be lost—according to Montluc, he
intended to stab himself, "which ancient Romans might do, but not good
Christians"—when St. Julian, the commander of his own Bulls, arrived
from the center of the battlefield and reported that the Imperial
Bulls there had broken formation after a long horn to horn tussle with
our own, and then been chased from the battlefield by our own
skirmishing hounds, which had been held in reserve.