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.
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.