Do you have the GM's permission already to use the custom race? Can we know more about it?
Also, is the archer a different PC or is he going to be played by you? Do you have permission to use Leadership if it's the latter?
"Standard stats" meaning like a human? Where you can choose where to put a +2, everything else is +0?
I would go with a sorcerer, with the elemental fire bloodline. Then you can use Beast Shape to polymorph into a mount.
Is there a reason YOU need to be the mount or could you just summon one? Or can you have a pet that's his mount all the time? If so then any of the three primary arcane casters would work fine for you. Same with shapeshifting, if you only need it for short duration spells can do that for you.
Fire Bomber Alchemist can turn into a Medium Elemental relatively soon, if your rider has Undersized Mount he can ride you.
Otherwise, the regular Alchemist has some fire stuff and Beast Shape in their spell list.
Bloodragers also have Beast Shape, but way too late. Arcanists can go Brown-Fur Transmuter and still pick up quite a few fire spells.
Unfortunately, the only class that combines wildshape and decent fire spellcasting is the druid. You could however build your character around produce flame (using metamagic such intensified or extend to keep it relevant at later level) and maximize your number of attack per round (I think you can either rapid shot OR two-weapon fight with flames as thrown weapon). This kind of build would have a different flavor and gameplay than a standard summoning or melee druid.
@ secret wizard : using a minute level spell like elemental body or beast shape isn't usefull to serve as a mount. Wild shape is hour level, which gives some true utility to serve as a mount.
You could go the lycanthropy route with the little guy and then he can be whatever class you want.
I thought he meant using it for like battle or something, where a couple of minutes are more than enough.
If he really really wanted Wild Shape... then you need Wild Shape.