Counter Strategy Alchemists
As they cannot see the future, they can be taken by suprise.
I assume the beasts they created have some kind of handler, or some other form of restraint which is released when they are deployed in battle.
Even if there are genetically engineered to be controlled by special trainers and otherwise leave the rest of their population in peace, this controls must be broken.
If the Chimeras can be roused to go on rampage in the Alchemists staging posts, outposts or even in their cities, soon they would either stop deploying them for the sake of their country or would start an all out assault, which might prove to be a costly mistake for them.
The Chimeras would be produced somewhere, or breed somehow.
As the Metallurgist apparently also conduct genetic experiments, if the genetic information used to build Chimeras can be obtained as a sample, it could be reverse engineered to send back their own Medicine to the Alchemists in form of Metallurgist Chimeras (perhaps with Exoskelleton Enhancement).
The samples could be even picked up from a Chimera on the battlefield, as all that is needed is a piece of it (even if its a flap of skin cut from its big toe, it still carries its whole genetic code) and a centrifuge in the labratory to isolate the DNA.
As their breeding program would have to be maintained developed and guided by specialists, subverting, kidnapping or assasinating those specialists would prove very beneficial in the struggle for supremacy in the realm.
Counter Strategy Scholars
As they can see the future, raids and betrayal wont work on them.
However, one doesnt have to stage an attack to gain the upper hand.
The secret lies in stealth, secrecy and sabotage.
Their airships are their biggest asset, and yet their biggest threat.
For one, if instead of an all-out attack, agents and spies are send into their airports, it is unlikely they will be detected from the seers.
They therefore would be able to either sabotage the airships, that they might not deploy or do a forced landing upon deployment.
Another target for sabotage would be the ammunition and bombs stored in the airships and in ammunition depots. If fire can be set to either one, the result will be a blazing inferno that might even spread into the city itself.
The advantage here is, that even when the seers are able to detect the impeding attack, without their fleet they will know just as well that they don't have anything to throw against it, rendering their skill useless.
Alternatively the Engineers who built develop and maintain the airships are vulnerable to attack.
If the lead researcher in their airship program is assassinated (which can be done with something as simple as a piece of cloth, twisted and with a knot in the middle to form a Garrot), it might throw them off balance.
Also, airships are hungry beasts when it comes to supplies, as the bigger they are, the more there is on them to break and the more high quality goods they need to keep them running.
Starting to attack, disrupt and destroy the production and delivery of for example the cloth they use to make the balloons that keep the airships afloat can reduce the deployment time of a fleet considerably, so that they will wreck havoc for shorter periods and over shorter ranges in ones own territory than initially.
Counter Counter Strategy Scholars
As the Scholars are able to see the future in form of impending attacks, the sabotage missions can be timed to coincide with another event: The attack of the Alchemists on the Scholars.
To be able to get to this point, one would either have to infiltrate the ranks of the Alchemists and make them believe that the Scholars have developed a new super-weapon able to undo the balance of forces in the realm (perhaps a dirty bomb or something).
If Infiltration or double agents are not successfull, the Metallurgist might come out and tell them, well, we're sorry for the experiments we're conducting on humans and animals to make something better (gently mentioning that their Chimeras are actually what they strive for), but however the Scholars definitely pose the bigger threat, due to their new super-weapon.
Then a temporary alliance should be formed, but that is not even necessary. It would only be required to make the Alchemists mistrustful enough to increase their surveillance of the Scholars territory.
Agents there could be used to conduct activities and drop clues (perhaps even a hijacked airship that is doing target practice with a new shiny bomb, whose powerful visual effects mask its uselessness) which would eventually make the Alchemists launch an attack.
Once that is known, the agents in the Scholars territory would have to activated, as by then the seers would definitely focus on the imminent (and real) threat of an incoming Alchemist attack and concentrate their air fleets on staging points and airports.
This way concentrated, they would be much easier to be taken out, as they are either on the ground, or docked and receiving maintenance.
The Metallurgist might even be able to sneak people onto their flagships which despite the sabotage might still be functioning, and murder the captain or sabotage the ship while in enemy territory so their captain falls to either the Metallurgist of the Alchemists, who would then have quite an upper hand in a bargain with the Scholars.
As in all wars, infantry would be involved to take land from the enemys forces and to guard it against them.
In ancient Japan, Shuriken were dug into the land around the fortress. Infiltrators and Samurai, usually only wearing sandals, would step into them and cut themselves.
The cut itself might not be fatal. The tetanus infection they would receive almost certainly will be.
Same goes for the projectiles hurled at the Alchemists Beasts and soldiers from both sides.
During European Medieval the usage of poisoned arrows was considered unknightly.
However British Longbowmen stuck their arrows in front of them in the ground when firing (for easier reload) hence infecting the once hit with Tetanus.
Alternatively a plant like Hemlock can be used to poison the arrows of the bullets.
If all that is not an option, and dirt on the bullets would make them misfire, one might just dunk them into poo. This will also ensure an infection after the projectile penetrated the enemy forces.
This would allow your bullets, which already can pierce the Chimeras outer layer, to penetrate deeper and deliver the payload of those deadly little helpers.
No Calculations necessary!
The game of treason with the two factions might be complicated, but as long as they don't get wise and launch an all-out assault against the Metallurgist, it eventually would roll their way.
Due to the possibility of an all-out attack, a group of infiltrators should be stationed in the Enemy Factions Capitals, to unleash Chimeras in the Alchemists from Control and set fire to Bomb Depots in the Seers realm, in order to have a counter strike in such an event.
Such an attack at any rate is more unlikely, due to the fact that committing considerable resources to attack an Enemy that is strong on the defence (The Metallurgist and their superior structures) would leave them themselves vulnerable for a strike from the other faction.
This is why it is so important to sow distrust and suspicion between the two other Factions, that they might never get wise and team up against oneself.
It is not recommended to take down all infrastructure and population centers in the other factions territory.
Doing so might yield a small advantage initially, like burning down a village here or razing a city and a fortification there.
But not stopping at that and rounding up and decimating the population, or destroying the life supporting infrastructure thus forcing them to flee, would fast backfire as the reputation of the Metallurgist as a bloodlusty pack of Barbarians would make them both much more unpopular in the land they try to conquer and it might also convince the two other parties to work together to do a joined attack against the metallurgist.