The gate isn't the problem, the guars are!
If the gatehouse is not guarded, breaking in is not hard. A portcullis can be simply jacked up with help of levers, a drawbridge can probably be unhooked from the chains if you can bring a ladder and climb up, a door can be chopped up by axes… The reason the attackers can't do that is that the defenders will be shooting at them, pouring boiling oil on them and doing any other nasty thing they can to keep them off the gate. So what you really need is to disable the guards.
1. If you can control pathogens, that's probably the most efficient option. Pathogens is the one thing medieval people have very little defence against.
Note that shooting dung and rotting carcasses into a castle with a trebuchet in the hope of starting a disease was a thing in sieges.
2. Another option is insects. A huge swarm of angry hornets is another thing the guards have very little defence against. They can hide inside, but they need some opening to shoot or pour hot oil on you and the hornets can get at them through that opening.
So if you have control over them to prevent attacking your own, they are better than covering fire. 548
They can allow your engineers to come to the gate and disassemble it in any suitable way.
3. Massive woodworm infestation can also do some good in a long lasting siege.
A siege means the attacking force blocks all access to the castle and waits until the defenders eat all the supplies and surrender because they are no longer able to sustain themselves. Which takes month to years. There is basically no fighting, everybody is just sitting around and waiting who runs out of supplies first (the attackers can bring supplies easily, but it's still expensive).
Infesting the wooden parts of the castle—the roof frames, furniture, wooden lining of rooms, barrels holding the supplies and such can make the castle uninhabitable, persuading the defenders to surrender sooner.
4. All large animal options are basically just getting cheaper labour.
Humans have dominated the Earth because humans with appropriate tool are better than any animal in almost anything they want. But soldiers have the disadvantage of demanding pay in addition to food, so if you can control some animals that only need feeding, you can make the operation cheaper, which means you can maintain the siege longer.
You can use some burrowing animals for undermining the walls (conclude with good old gunpowder explosions one the burrows are dug), and you can use whatever dangerous animals as guards on the paths to the castle for maintaining the siege.
What is most dangerous depends on the precision of control you've got. A chimpanzee with a spear would be significantly more efficient than a tiger. A tiger has short reach so a human soldier with a spear can pierce it before the tiger can touch the soldier, but if a chimpanzee can wield the spear with some efficiency, it has the same reach and more strength in a smaller package. 164