Answered in depth by Cal, but again spoilers within: [SPOILER]Remember that Louis became a Protector towards the end of 'Children. Louis relied on the other two Protectors running the Ringworld, Tunesmith and Persephone, believing that Louis would be doing everything possible to hide his son, Wembleth, from them.
This part of the wiki discussion on Pak Protectors though suggests that both Tunesmith and Persephone in actuality weren’t as driven by the bloodline preservation instinct as (according to Brennan, in Protector.) normal Pak Protectors were.
Anyway, the ruse wouldn’t work if either protector didn’t believe that Wembleth wasn’t actually Louis’s son. I forget, but don’t both Tunesmith and Persephone find traces of Wembleth and Roxanny, as they’re chasing Louis through the stepping disk network, smell them, and confirm that Wembleth was his kid?[/SPOILER]
And luck (at least, Teela’s luck) doesn’t work that way, anyway. Human luck is described as being an evolved trait, in response to environmental pressures implemented by the Puppeteers. In other words, to be lucky doesn’t mean to get what you want. It means to have lots and lots of descendants. Teela becoming a sterile Protector without first having any children, or only one child, is a phenomenal failure of luck, not a success.
Well if you have one child and save the Ring (so that one child can have billions of descendants) that’s better than having lots of descendants who all die with the Ring after a few decades, so in a situation where those are the only choices “lucky” might well be to become a Protector with only one child.