We were the only force outside the city wall/gates from what I could tell so presumably no one else was able to desert during this battle, yet.
If I was sieging a place I'd have my men kill any messenger birds that get sent out. That's if there is even anyone for Fallory to call for help from, it is the capital after all, it possible that the invaders have already conquered the rest of the nation. I've got nothing to say about magical means or attempting to slip out a messenger, those are both probably possibilities.
The massive invading army is gonna be kinda busy sieging the capital for the next 6 months-1 year to start going around boasting about getting us to leave.
Kill every single bird? Including ones released at night, ones that are flying high above the walls? That's potentially a tall task. The defenders had the benefit of knowing that a bird would be coming with an offer ahead of time, knowing the specific direction it would be coming from (the side of their walls facing the giant oncoming army) and the light of day when they torched that single bird. And it was flying close to them, as it was trying to get to them over the walls which the defenders are at the top of, so that allows for the distance the ranged attacks must travel to be minimized The besiegers have none of those advantages, and again it's a six month long period. People can certainly grow lax even if they were given such an order.
It doesn't even have to be official communications demanding troops be sent here and there. People have families and loved ones, you know? People who they might want to tell about how the city is being put to siege, and how that came to pass. Rumors spread faster than men... and as you admit, messengers or magical contact are likely possibilities.
How do you think a siege is supported, also? Unless they're magically conjuring food out of thin air, they've got to get supplies from somewhere. They'll be bringing some of that from home, but oftentimes that is supplemented or even outweighed by local foraging. That's hunting and the like, sure, but more prominently also in getting food from the local peasantry. Whether that's done through purchasing, intimidation or outright theft and robbery is generally up to the local commanders, but when you have a massive army they're going to have some contact with the local populace. They may very well have a bunch of camp followers from this kingdom as well. Do you think that none of them will ever talk to the surrounding population about the outcome of this battle?
That's why building a reputation as a mercenary company is a good idea. Spread rumors of our own that we were never present at the battle, build up a good reputation before rumors can seriously start to spread and that reputation will shoot down any rumors about us leaving the battle of Fallory. We go the bandit rout and when the invading army finishes their conquest they are now free to focus their efforts on killing our sorry asses. We go mercenary that doesn't happen unless we decided to provoke them first.
EDIT: Hell failing getting work from the natives the invaders will probably be willing to hire us.
Just spread a bunch of lies about it? Now, how much do you think anyone else will value this when we're clearly members of the Nameless, and were fighting this threat all throughout the kingdom? What will they think about that when we are out requesting pay for whatever work while the capitol is under siege? It's not like anyone would have valued our lives enough to release us of our duty and surrender a fraction of their desperately needed army.
Banditry follows a similar rule; don't try to rob those in power. I don't think the invaders particularly care about the peasants out here, and they certainly don't have full control now. We can avoid raiding the towns that they control when it comes to that, if it comes to that. We don't actually know what they're out for, it may not be outright conquest.
And for that matter, I think any mercenary work that we could get would mostly revolve around protecting totally desperate people (the only sort willing to overlook our flaws) from the only army standing's foraging parties... which would obviously irritate them. The invaders have no need for men especially of our dubious character, they have enough mouths to feed as it is.
