Step 1: Re-center yourself.
You've been engaging in a rebel campaign to unseat the king. The king is now striking back. This is the time to re-evaluate your priorities. Even mercenaries have to re-evaluate their priorities from time to time:
Priority number one: get paid.
Priority number two: Live long enough to spend your money.
- Tagon, Schlock Mercenary
The king has now made it his focus to remove you from existence. How badly do you want to try to depose him? Time to refocus. Your goal is to survive the onslaught while retaining that essential essence that made you devoted to your rebel cause in the first place.
You will need to engage in vastly asymmetric warfare. Your opponent has overwhelming physical might. You will need to win the battle on the mental front, encouraging him to decide to spend his resources elsewhere. Once he is no longer focused on your eradication, you can resume the traditional "short live the king" mentality.
Step 2: Patience.
This is the hard part. Your opponent has overwhelming force and wants a decisive victory. You may want a decisive victory too, but you're not going to get it. That's why we invented swords and armor... to make sure the other guy doesn't get a decisive victory. This is going to be a long fight. You need to make sure that every resource you spend is spent worthwhile.
The best approach to this is to Stop Worrying and Learn to Love the Bomb, I mean... sorry... wrong storyline for a moment... where was I.
The best approach is to turn the fight into the kind of fight where you fight them simply by living. That way, no matter how the fight goes, you went on living a good life. This is going to involve some creativity. You are going to have to find ways to make the presence of the king's men a beneficial part of your life, despite all their pointy sharp things.
This will take time. An army is usually not structured to make the enemy's life cushy. You're going to have to calmly and patiently explore their structure. There wont be a one-size-fits-all solution for you, or the army would stop it. You have to look at how you and the enemy army will interact, and find some details about that army and about your men which let you live. It's all up to your imagination here.
Step 3a: Vanish
Don't worry, I wont leave you hanging with some whims of imagination there. While there is no one-size-fits-all solution, there are directions worth looking in. The most important step is to live long enough to get paid... or...well... whatever you're rebeling for. Your opponent is designed to build a net to capture a band of merry men. You need to cease to fit their mental image of what a band of merry men must look like. You need to become so completely one with nature that, when they send out scouts to draw the net around your band, you can simply flow through the net like water through a sieve.
There are some tricks here. This is war after all. I would quickly train any animals like foxes, falcons, and wolves to go after the dogs the army brings along. Your human smell is your biggest weakness. Dogs can sniff it out from a mile away. However, trained dogs are not cheap. It will strain their resources if you can pick them off one by one. You may have to be creative here, like in every step. You may have to structure your own scent trail to draw dogs into kill zones. In all, you want the battle of the nose to work in your favor. Once the dogs are neutralized, you are just fighting men. Men are not as perceptive as dogs, so it will be much easier to vanish.
Step 3b: Recon.
Train your animals to give you information about where the enemies are. The birds are already trained. They're going to give you tons of information about where they are. While you're at it, return to step 3a and train yourself and the birds so that the birds don't give you away. Become one with nature.
You are fighting the mental game. You cannot fight the mental game without insight into where the enemy's physical force is. Wherever their physical force is, that's where you need to not be.
Step 3c: Supply lines
If you retreat backwards into the woods, you will force the army to develop supply lines. These are known to be a great weakness of armies. If they are sparse enough to live off of the land, they may be picked off one by one. That's not the army's style, they're more likely to draw a supply line from the kingdom to ensure they have supplies of men, food, and possibly more dogs (you'll have to keep picking them off).
Supply lines are great, because they're ripe with things you want, like swords, food, and more swords (I mean food, swords, and more food!). Best yet, they are long and remarkably hard to defend against unless you know where the enemy is (you did remember to vanish, right?). Send in the racoons to wreck havoc on their supplies. Maybe you can even teach the greedy bastards to bring back some of the food they steal!
This is where the cost mounts for the king. Supplying an army is not cheap. Doing so with losses along the supply lines is even more expensive. This is where you get to send your mental message: we are less bothersome as a band of rebel merry men than as your mortal enemy. If you stop treating us as a mortal enemy, we'll go back to just being pesky and ineffectively trying to depose you. Convince him it's too expensive to crush you, and he'll eventually have to back out. Or, if he's really a jerk, he'll raise taxes, but hey look... easy way to piss off a kingdom enough to remove his cushy butt from the throne (or was that "butt from his cushy throne?" Those adjectives are easy to misplace). Either way, the war will move in your favor.
Step 4: Bear.
Bears are scary. Bears are mean, nasty, and capable of gutting a man like a fish by accident if they hug him too hard. Did I mention they are scary?
Did I mention they are even scarier when you are an isolated army with inadequate supply lines, worn out weapons, and a really creepy spooky forest full of violent trained animals, at night?
If you do a really good job of training the bear, you can even convince it to leave a few men behind to tell stories at the tavern of just how huge the bear was with glowing red eyes and smoke coming from his mouth.
Then you can go back to rebeling to your hearts content. The king may send an army to attack a band of merry men, but a mythical demon-bear-beast worthy of a a place in Norse mythology? Fat chance.