I was wondering if they would also be eligible for residency beyond the initial 3 months
Any citizen of an EU member state has the right to live here in Poland. The three months thing is an irrelevance - basically a legacy of the pre-EU system. They can't make an EU citizen move elsewhere, nor can they enforce any conditions on your stay. The residence documents are not a permit, nor can you be asked to leave if you choose not to apply for one.
You may choose to register residence, however whether you do or whether you don't, they won't deport your baby!