My understanding is that your best bet may be to have your company petition for an exception on your behalf. Without a child, your partner could probably find work here (SG has very low unemployment, it seems like foreigners willing to accept local compensation packages are competitive). If your daughter were legally tied only to you, you might be able to secure a dependent pass for her as if you were a single parent. But assuming you have only a joint adoption decree and that Singapore would require the involvement/consent of both parents before granting a child a visa, then your paperwork limits your ability to work around the system. Does anyone have a sense of how the SG authorities would look on an adoption decree with two men as parents? Is it an option to accept the position contingent up on visa approval for your family?
I have heard about foreign children being eligible for student visas even at the preschool level and that a child enrolled in certain MOE-approved programs can sponsor a parent's visa. Apparently this is sometimes done by wealthy folks in the region who want their kids brought up in an English-speaking school system. Again, no idea whether a kid could get enrolled and approved with two dads...
When my partner and I asked around about visa options for same-sex couples, we heard rumors of SG making exceptions for senior leadership at MNCs that the country wanted to court. But the knowledgable people we met with informally were reluctant to suggest we try the same thing (my partner is senior but still several steps away from the C-suite). They did all agree that legal marriage would help our case.
One of the challenges we encountered was that Singapore doesn't really have immigration lawyers as many other countries do, so we couldn't locate a specialist on the topic, despite having a handful of local friends and allies.
The other issue to consider, of course, is being a two-dad family in Singapore. Depending on your daughter's age and school options, it could be less than ideal for her. I only know one same-sex couple in SG with kids, but they're quite jaded. On the other hand, if your kid can be insulated in a friendly
International School (which would that be?), it might be okay and the experience of living abroad would certainly be valuable for her (assuming you're not already expats).
Good luck!