Yes.
This is one of those classic "already-been-solved" problems:
Mice in the wild face existential threats every day. They've already figured out their best strategy: find very safe places to hide, and scurry among them when necessary.
I don't see why this would be any different in a D&D-verse. The urchin's mouse--during fabled background times--will have figured out the good places to ride along. Wedged between a waterskin and spare jerkin in the pack? Between layers of a tied bedroll? We may find those locations claustrophobic, but that's comfort to a mouse.
Even if combat kills your poor urchin, I'm sure the mouse could be fine. Unless caught out in the open. And that's the life of a mouse.
There was much made in comments of whether the mouse could possibly survive the hypothetical fireball in OP. Please see Would a very small creature inside a sealed container carried by a character take damage from AOE spells? for the correct explanation of how it could.
Paladins are not Druids
The Oath of the Ancients has nothing to do with nature; it is about life and light.
A Green Knight has committed to fighting for a world where things can live and grow and flourish. Here, "things" includes plants and animals, true, but also includes people and ideas and art and concepts and societies.
Green Knights use growing things as their symbols because those things are the opposite of death and decay, which symbolise what the Knight is fighting against.
They do not necessarily have to kill wicked people and monsters. A Green Knight's focus is providing an environment where the light can prosper. If that can be done without killing (perhaps by moving the creatures on), then that is OK (though individual Paladins may differ on the level of restraint they feel is acceptable). If the evil creatures can be redeemed and brought over to the side of the light, that is (in my opinion) the best outcome. (Hat-tip Molot.)
Examples
Travel with evil aligned characters without any attempt at reforming them?
No. Green Knights are supposed to stand against wickedness, not stand beside it.
If there is a greater evil, the Knight might put up with a lesser evil, but only temporarily. Even so, the Knight should at all times try to redeem the wicked, through the example of their own actions (Be the Light).
Forego forgiveness in favor of killing someone who betrayed him?
Maybe - there should always be a hope of redemption. However, a strict Knight might feel that one transgression is one too many. Any Knight might refuse forgiveness to one who has betrayed them several times.
Overlook the evil and wrongdoings of man? (or any sentient race)
No. Again, the Knight is supposed to stand against wickedness, not stand aside.
They are not, however, stupid. They won't engage in pointless attacks. Throwing your life away for nothing violates the tenet of Preserve Your Own Light.
Inflict injury on someone for lying to him?
Maybe. A very strict Knight might be justified in punishing lying with injury.
Media Figures
I think Wonder Woman is an example of a Green Knight.
From an interview with Patty Jenkins, the director:
Jenkins goes on to describe Wonder Woman’s worldview as, “I believe in the betterment of you, and I believe in you, and I believe in truth. And I believe that you all can be better, and I will fight to protect you. But I also believe in better than that.”
Best Answer
Don't get hit
The best protection for a low hp, low ac creature is not being targetable. Keep your falcon high in the air, behind cover (but not the same cover as you or your allies, for reasons below), or otherwise not in or near combat.
Problems with Armour
First off, by RAW barding costs 4x and weighs 2x of regular armour of the same material. (PHB p. 155) Personally, it doesn't make sense to me to make armour for a tiny creature weigh more than for a medium creature. The price could be explained as being more difficult to create the armour, but I assume a large chunk of that cost increase would be materials, so talk with your DM about some house rules on falcon armour for more sensical or feasible costs and weight.
Another note, is that falcons are not inherently proficient in any armour. This is entirely up to your DMs judgement, as I don't believe even Warhorses are called out explicitly as proficient in armour.
Even with barding that would make the falcon have a ridiculously high AC, he'll still be hit 5% of the time with a natural 20. With the low HP of the falcon, and the ever increasing damage that monsters and npcs will deal out, he will likely be knocked out or killed if he's hit.
As for flight, I would likely house rule that a falcon can only have light armour at most, as this is the same limitation that the Aarakocra race has on being able to fly in armour.
AoEs and You
Another threat are AoE spells, which a kind DM would let you protect the Falcon while he's on your person, by RAW it is a separate and targetable creature. Nearly any AoE spell will at least knock out the falcon, and even a low roll on a Fireball with a Save on the falcon's part has the potential to kill him.
Note this also does not preclude the use of spells directly at your falcon, such as Magic Missile.
Alternatives
Find Familiar
Take the Magic Initiate feat (either from Variant Human, or your level four feat), and choose Wizard for the Find Familiar spell. You also get two wizard cantrips out of this. (For a Paladin Green Flame Blade or Booming Blade are both nice, as well as other flavourful options.)
A familiar from this spell can still die, but you can bring it back for a mere 10 gp and an hour ritual. You also get several other benefits, such as telepathy, seeing through its eyes, and being able to be safely dismissed to a pocket dimension during combat.
Talking to your DM
The only other real way around your falcon's mortality is talking to your DM for an arrangement to make the falcon not killable. Everything else in terms of items, barring an artifact of immortality or similar, would only make the inevitable slightly less likely.