Sorry if I sound mean. But when you state things as facts - like you KNOW that your child will be a UK citizen when it won't...well, I feel the need to correct you. If correction = mean, then I apologise again.
From your own link:
How can my visitor come to the United Kingdom?
An entry clearance officer must be satisfied that your visitor meets the requirements of the Immigration Rules, which are that:
he or she is seeking entry as a visitor for no more than six months and
he or she intends to leave the United Kingdom on completion of his or her visit and
he or she has enough money to maintain and accommodate himself or herself without working, or help from Public funds (income support, housing benefit, etc.)
And, yes, I underlined that bit to point it out.
Also
This page is primarily for someone who is sponsoring a visa national, but will also be useful if you want to sponsor a visitor who is not a visa national. You may also want to read the Sponsors (INF 3) guidance note which gives more detail.
BTW - you as a US citizen are not considered a visa national, just in case you were wondering.
And Sponsors (INF3) guidance note says, again
How can my visitor come to the UK?
Your visitor must show that:
they want to visit the UK for no more than six months
they intend to leave the UK at the end of their visit, and
they have enough money to support themselves and live in the UK without working or needing any help from public funds
and
How long can my visitor stay?
Your visitor can stay in the UK for up to six months each visit. A visit visa gives the person permission to enter the UK and allows them to visit the UK as often as they like while their visa is still valid.
If your visitor travels to the UK regularly, they can apply for a visa that is valid for up to 10 years, but they can only stay for up to six months on each visit.
This visa is NOT for someone whose intention it is to ultimately settle in the UK.
And if trying to make sure someone else doesn't suffer the embarrassment of being REFUSED ENTRY when they feel they are doing things that are *technically* legal is mean - then I guess I'll just bow out of this thread and let you continue on your way.
Just seen too many people coming here for help, getting good, clear, concise advice which they decide they don't want to follow, and then coming back asking how to fix the mess they've gotten themselves into.