I had a less mechanical option. I told my players if any of them were interested in mounts that I would make sure there would be available increasingly fantastic creatures. Yes, keeping your warhorse is cool, but getting flying mounts or unicorns or the like is also cool. Plan was to offer CRs the same as a Moon Druid - character level/3. That's powerful enough to survive and contribute without eclipsing other characters who are not taking a mount option nor characters who are spending class features on it - a real balancing point.
Also if you go about it via the Companion's Bond route, Barding would just be adjusted to follow shield rules and apply a +1/+2/+3 to ac for simplicity's sake. Of course, you would still keep the increased cost the same just for the sake of not being seen as completely insane by your table.
But I would make it a feat.
Just so there is some investment.
requires proficiency in animal handling.
+1 to any ability.
Pick any beast of your size or one size larger.
You can use that beast as your mount.
HP of the beast is 5 + 5×your level HP.
Beast uses your proficiency bonus or it's, whatever is higher.
beast has evasion and proficiency in all saves
beast adds your PB to it's AC
when you gain ASI, so does your beast. Your DM might allow your beast to take some feats instead of ASI if they are appropriate for the beast.
(Mobile, though, lightly/moderately/heavily armored, Alert,)
During a long rest you can bind a new beast to you.
You lose your connection to current companion if you have any.
As with any NPC that takes part in combat on the side of the PCs, I give combat mounts a share of XP, although they only receive a half-share as in 1E. When they have enough XP to level up, they can add class levels according to the rules on DMG p283.