It’s je me suis pincé. As soon as you use a reflexive pronoun, it has to follow the normal conjugation rules.

All verbs that use the reflexive pronoun are pronominal, and all use être for the auxiliary in compound tenses.

Ad-hoc pronominals like se pincer can be formed with pretty much any transitive verb and function the same grammatically.

The reason a pronominal verb appears in the dictionary is usually that its meaning is not completely transparent given the meaning of the base verb, or it has an idiomatic sense on top of the literal one. So it’s a semantic distinction, not a grammatical one.

In addition to the already existing excellent answers, maybe does your confusion come from the definition of pronominal.

Pronominal doesn’t just mean that a pronoun is used along the verb, but that this pronoun represent the subject (i.e. is a reflexive pronoun).

When it is not the case, the pronoun is just a regular complement and the auxiliary is avoir with most verbs, including pincer and demander:

J’ai pincé "toi" → Je t’ai pincé.

J’ai demandé "à vous" → Je vous ai demandé.

On the other hand, when the pronoun is also the subject, only être can be used:

J’ai pincé "moi" → Je me suis pincé.

Vous avez demandé "à vous" → Vous vous êtes demandé.