In this case, I think the naturalisation certificate (not passport, which is not definitive proof of British citizenship but merely a travel document) is sufficient.
The ILR BRP or proof of ILR is only relevant in case the child is born in the UK after ILR but before British citizenship.
Registration under Section 1(3) requires one parent of a child born in the UK to be either a person settled in the UK or a British citizen.
Section 1 of the British nationality Act 1981 wrote:(3) A person born in the United Kingdom after commencement who is not a British citizen by virtue of subsection (1), (1A) or (2) or section 10A shall be entitled to be registered as a British citizen if, while he is a minor—
(a)his father or mother becomes a British citizen or becomes settled in the United Kingdom; and
(b)an application is made for his registration as a British citizen.
As for good character, yes, your child will need to meet the good character requirement. See the
good character caseworker guidance.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.