The simplest reading should always be the first consideration: and in this case that implies Keturah was married after Sarah's death, and the multiple children came at the end of Abraham's life.
This straight forward interpretation is upheld by other text considerations, like: There are no other children or wives mentioned in the events surrounding the circumcision, sacrifice of Isaac, kicking out of Ishmael & Hagar (only), "let Ishmael stand before you!", etc.
So the question seems to fundamentally come down to the interpretation of the Hebrews passage "one as good as dead". Most commentators seemingly blindly assume this means Abraham was unable to procreate. But this does not have to be assumed.
The texts show that Sarah is clearly barren, and a miracle was required to open her long-closed womb. But there is nothing in the greater texts to warrant the assumption that Abraham was also barren, outside this one verse in Hebrews. It makes little sense to assert that Abraham was later miraculously made fertile, when he had a son by Hagar before the later visitation, Keturah's obvious fertility, and that ~90 years for a man at that time was not at all old physically when we see the large number of children born in Genesis to much-older men.
Instead, I assume the Hebrews passage about "good as dead" is an assertion about Abraham's (and thus any man's) inability to procreate a spiritual birth or inheritance, and not directly about a physical birth. This seems to be an overarching theme of the whole context of Hebrews. Galatians 4 (vs 21-31) also focuses on the comparison of the two children, one of a promise and the other of a physical birth. Note that the other children of Abraham are not mentioned in Galatians 4, nor is Keturah and her children. It is a contrast of physical & spiritual. Abraham was clearly unable to bring about the fulfillment of the promised child through Sarah.
So then a spiritual birth must be the meaning in the Hebrews passage, instead of about Abraham's physical fertility. And then we have no reason to accept any other answer but that Abraham married Keturah after Sarah's death.