Pronominal verbs use exclusively être as an auxiliary verb for composed tenses. Therefore, independently of the word order, your first phrasing can't be correct.
Your second guess was right. To augment your level of confidence in it, you might want to proceed step by step. To avoid potential confusion due to homophones “est” and “ai”, best is to also consider the 1st person.
Start with present indicative for the auxiliary verb:
Je me suis réveillé.
Il s'est réveillé.
Add negation:
Je ne me suis pas réveillé.
Il ne s'est pas réveillé.
Turn this into question form, with subject–verb inversion:
Ne me suis-je pas réveillé ?
Ne s'est-il pas réveillé ?
Switch to conditional:
Ne me serais-je pas réveillé ?
Ne se serait-il pas réveillé ?
You may of course consider different paths. Yet, your final guess was right.