Thank you everyone for your inputs on this. I went to DSWD and top-secret was right. We do need a consent signed by both grandparents along with some legal documents from the mother's side (mother's birth certificate, death certificate and CENOMAR, grandparent's marriage contract and valid IDs) plus all letters from USCIS and NVC and copies of some visa requirements (petitioner's passport, either affidavit of support, COE or bank statement), as per instruction. DSWD will also need to interview the grandparents.
I wrote down the requirements just for the sake of this thread. Someone with the same case might need this information.
Seems like it's a lot of work compared to an illegitimate minor child traveling with their mother. Anyhow, I'm going to gather the requirements and comply with DSWD.
Thank you Ontarkie, Chancy and top_secret for your help.