Proving that "at" is a preposition in "he laughed at me"

Laugh at, as Bill Franke tells us, is a “phrasal verb” — that is, a construction of the form V + P (Verb + Preposition) which must be treated as a single term with a meaning distinct from the use of the two components in merely accidental juxtaposition.

For example:

In I laughed at sunrise, laugh is the ordinary intransitive verb meaning “to express mirth”, and at sunrise is an ordinary prepositional phrase telling when I laughed.
In I laughed at the sunrise, laugh at is a transitive phrasal verb meaning “to mock”, and the sunrise is its object.

There are two sorts of phrasal verb. In the first sort, the P-component must appear before the object of the verb, so it looks exactly a preposition heading a prepositional phrase. Laugh at is phrasal verb of this type; we say

John laughed at Bill’s stupid mistake. ... but we do not say
*John laughed Bill’s stupid mistake at.

Linguists tend (but usage varies) to call the P-component here a preposition.

In the second sort of phrasal verb the P-component may appear (but is not required to) after the object. For example, we may say either

John ran over the pedestrian. ... OR
John ran the pedestrian over.

In this case, linguists tend to call the P-component a particle.

Where the particle is placed in these phrasal verbs tends to depend on the “weight” of the object: the longer the NP which acts as the object of the verb, the more likely it is that the particle will be placed before it:

John ran over the pedestrian in his path is more likely than
^?John ran the pedestrian in his path over.

Accordingly, the best “test” for whether the P-component in a phrasal verb is a particle or a preposition is to offer a Native Informant a sentence with the “lightest” possible object — a simple pronoun — preceding the P-component and ask whether that sentence is idiomatic:

^?I laughed him at.

However, there’s still one tricky bit here: some phrasal verbs have two meanings, distinguished by particular or prepositional use. For instance, run over may mean either “knock down and tread upon in running” or “examine from beginning to end”:

John ran over the pedestrian in his path ... or
John ran over the list of alternatives line by line.

So when you perform your test you must make sure you provide a context:

John failed to see the pedestrian in his path and ran him over. ... acceptable
*John took up the list of candidates and ran it over line by line. ... not acceptable

A preposition doesn't have to be replaceable by other prepositions to qualify as a preposition itself. At is only a preposition, and in many cases it is the only preposition that can be used in particular constructions to express a specific meaning.

Still, in your sentence, at is not the only possibility.

He laughed with me.

He laughed for me.

He laughed near me.

He laughed by me.

And so on would all work, though they would mean different things.

Laugh at is a prepositional verb, and at is a preposition. Unlike transitive phrasal verbs, prepositional verbs do not allow particle movement (the particle in a prepositional verb being a preposition). The preposition ‘always comes before the noun phrase that is the object (‘Longman Student Grammar of Spoken and Written English’). We can’t say *He laughed me at.