One needs to make certain assumptions about the behaviour of the enemy.
At the moment, it is looking bleak for the people of Lusom: If the enemy truly has them surrounded, they cannot escape. If the enemy truly has technological and financial superiority, they would have no troubles destroying Lusom utterly.
If the enemy truly intended to wipe out Lusom, they would. And as a last resort, they could. However, Big Dog is only willing to get involved if, and only if, this will be an easy win.
I am led to the conclusion that it is important for your enemy not only to quell the aggressors within Lusom, but also either to a) keep Lusom civilian casualties to a minimum; and/or b) preserve Lusom infrastructure.
With that in mind, even though this is clearly a popular display of power by Big Dog, it is not a popular seige. Furthermore, the presence of mercenaries in a vast and capable army suggests an attempt to intimidate; Mercenaries would not risk their own lives on the front line except for considerable financial compensation, in which case state soldiers would suffice, and be cheaper.
Next, we should aim to understand the role of the allies of Big Dog. These are your political and geographical rivals. They have invited a force, far superior to their own, to fight on their behalf. Maybe they want you truly dead. Maybe they're already talking about how they're going to carve up the spoils of war.
If anybody is going to end up on the front lines against you, it is your rivals. Big Dog would not risk his own soldiers first, it is not his quarrel. He is no mercenary. He is a Lion amongst Hyenas. Or, a big dog type thing amongst several smaller dog type things.
So with these considerations in place, I'd consider these resolutions to the following scenarios.
S1 & S2) Prepare for Defence:
S1) Big Dog wants you dead, and doesn't need to send armed personelle into Lusom.
R1) You're screwed. (Napalm, Drones, Missiles, AC130s, etc, or Destroying literally all your resources and then just sitting there for 2 years so that when they eventually go home you've got nothing going on).
S2) Big Dog really really wants you dead, and needs to send armed personelle into Lusom.
R2) Guerilla warfare is your best bet. 300,000 soldiers among 3,000,000 civs. Provided you can protect your supplies. Just make it strategically exhausting for them.
S3 & S4) Incite discontent:
S3) Big Dog doesn't really want you dead.
R3) Demoralise all factions of the enemy. Target mercenaries, so they will leave. Target the rival powers forces, so they will implore Big Dog to fight for them. This would affect public support.
S4) Big Dog just wants to intimidate you.
R4) Continue for as long as possible as though nothing is happening. Incite skirmishes from the enemy, make them unfavourable.
S5 & S6) Modify the Elements of Risk/Reward.
S5) Big Dog wants your town, doesn't care about you.
R5) Shoot yourself in the foot. Irradiate the place with uranium or something.
S6) Big Dog is just trying to maintain the power balance.
R6) Convince Big Dog that they would be better off with you as an ally, as opposed to your political rivals, who are now extremely vulnerable because of your combined military presence.
One possibility which occurs to me, is a kind of reverse trojan horse scenario, where you can incite the enemy to send an investigatory force into the city, and you take hostages. If the hostages are valuable, you might be able to dissuade Big Dog from doing anything too insane to the city, like chemical weapons.
I also might possibly suggest that you declare your super bunker to be an incredibly large nuclear weapon which you'd be prepared to set off as soon as you've taken too many casualties, and which will blow up if it sustains too much damage.