I've found it.
Here is the fix in case someone else encounters this.
I opened TES5Edit and looked at the dialog records for Falk related to furnishing Proudspire.
The record for purchasing the rooms had no conflicts, only Skyrim.esm and Hearthfires.esm showed up.
But the topic:
DecorateSolitudeElse [DIAL:000E24D3]
Hearthfires.esm over rides Skyrim.esm to add the option to buy the children's room:
DecorateSolitudeChildBedroom "Children's bedroom. ( gold)" [DIAL:08003D4B]
I looked at [DIAL:08003D4B] and Unofficial Skyrim Patch.esp was over riding Hearthfires.esm's entry to buy the children's room (Unofficial Skyrim Patch.esp was omitting the option to buy the children's room.)
I moved Unofficial Skyrim Patch.esp above Hearthfires.esm in my load order, and now the option shows up.
Two things:
1) Is this a bug in Unofficial Skyrim Patch.esp? Why is it over riding that dialog?
2) I thought the game loaded all the .esm files first, then the .esp files. But when I moved Unofficial Skyrim Patch.esp above Hearthfires.esm, the error was corrected. What is the rule for loading .esp files before .esm files?
Thanks for all the replies and suggestions people!
I appreciate it.