Welcome to SEA! :) To address your two questions:
1. Most large global consultancies actually have a presence (and office) in both Malaysia and Singapore. These include the likes of MBB, OW, ATK, LEK etc. To my knowledge, several boutique strategy consulting firms, such as QVARTZ, AlphaBeta, Gingko Consulting, Lizana & Company, Eden Strategy Institute etc., also have a strong presence in this region. It would really depend on whether you have a specific interest in any industry / specialization, and whether you would be keen in joining some great regional boutiques or would prefer stick to the larger-known brands!
2. I found this forum / list which could be helpful in your research: https://www.spencertom.com/hub/topic/426-consulting-firms-in-singapore/.
Also, as a caveat, the suggestions above are mainly based off my observations as someone living in Singapore :) Perhaps someone who has experiences working in Malaysia's consulting scene could share more relevant insights to your current situation!
Wishing you a great time in Malaysia~