Premise: I've studied latin for about 5 years, so I'm not going to use gTranslate for Latin :)

my dictionary sounds better for this scope.

• Verb:

The verb prodeo [pro-eo] is the best I could think.

• the particle pro- means something like in front of, even between (me) and something, or near (me): this last one is peculiar and happens only sometimes.

• eo is the most common and even one of the ancient verbs (that's why is defective/irregular) that means simply go.

So when Cicero (Br. 39) said :

prodire in lucem

he probably meant something like

come out from dark in the light.

Other times is used, like in Caesar (Bg):

in proelium prodire

that should sound like

come out and go to the battle.

OT:
(If you need for other occasion for a "incantation" in a more militar situation , a good one could be subject in ablative case+ proelium proditu (prò-e-li-um prò-di-tu), but that's another topic :) )

Even flowers prodent and in a figurative way, even

lacrimae de gaudio prodeunt

(Apuleio)

tears of joy appeared/came out of (him)

but this sense doesn't matter that much for our problem, i guess.

• Subject:

Since you don't use "Hat" for the Sorting Hat, but it seems to me that you want to stress the fact that this "entity" is that important because is a Sorter I would guess

Deligitor

would be the best.

Also Eligitor would be nice there is a subtle difference: the last one means "the one who choose what he prefears". Deligo[de-lego -> de-eligo] means choose what (or who) is more apt to a peculiar aim.

A Cicero's quote:

ex civitate in senatum delecti

meant something like

choosen among the cives/citizens to form the senate(to be senators)

Deligitor is the noun formed by the verb, means "who choose, who looks for the fittest men (or stuff) for a task and choose them for that task"

• The spell: JKR spells are really more naive, but that's not the point: they are not meant to be real Latin, but they are meant to sound like "Ancient powerful spell with complicated and forgotten words", imho.

• The best grammatical looking phrase would be:

Deligitor prodi

that means "Chooser, be present"

because the verb is in the imperative mode, second person singular: prod-i. But that doesn't sound that good, imho.

A more free construction could be

Deligitor prode

That literally means "(the)Chooser has come to be present here."

and could be quite nice (not too far from Latin, not too boring for a fan-fic).

Or dozen of combination of this ones: deligit[or;-us,-um] prod[i,it,es,

oh! that could be nice also:

deligitor prodeas

is exhortative(or exhorting, i don't know) conjunctive, that simply means:

please, do this or would you mind doing this or it could be perfect/awesome if you bother to do this

that would sound like Chooser, please come here asap , or Sorting hat, come here since we need you

I'll stop here, waiting for some feedback, because otherwise my mind would be lost in this long trip.