Learn Spanish (Latin America): Él no me deja recordar el truco.

In causative structures (using verbs like hacer or dejar to cause or allow an action), the pronoun representing the person being allowed or forced to do something usually goes before the conjugated causative verb. Placing the pronoun before deja is the standard, natural way native speakers build this structure.

Is me a direct or indirect object pronoun here?

It acts as an indirect object. In causative structures with dejar and hacer, if the second verb (recordar) has a direct object of its own (el truco), the person being allowed to do the action becomes the indirect object. Because me looks the same for both, it doesn't visibly change here, but if the sentence were about letting 'her' remember the trick, it would use the indirect object pronoun: Él no le deja recordar el truco.

Could I use acordar instead of recordar to mean 'remember' here?

You could, but you would need to use the pronominal form acordarse, plus the preposition de. The sentence would change to: Él no me deja acordarme del truco. Notice that a second me must attach to the infinitive to match the person remembering. Because recordar takes a direct object without needing a preposition or a reflexive pronoun, it keeps this causative structure much simpler!