If Harry decides to live in the US permanently (green card, etc), he will have to pay tax on his worldwide income, as has been said previously.
He can in theory work around being a permanent resident and tax thing by good tax advice and BUT ALSO by being out of the country for certain periods of time. However, it is complicated because the number of days you are allowed in the US as a foreign national does not restart from a particular date and run for a period of a year (Jan to Jan.) If you spend a certain number of days in the US one year - like if you max out the number of allowed days - the number of days you are allowed to spend there the following year goes down.
It's very complicated and a tough way to manage family life. I do not pretend to know all of the ins and outs but I only have a window into this from a dear friend who is also an expat but whose husband was not a US national (similarly minted to Harry, in fact, although not his Dad). She talked me through this during one of the periods I was thinking of renouncing. She was desperate to go back - nearly ended the marriage tbh. Not a chance did he want to be a US green card holder. For a little while, they came up with a solution to spend half the school year in one country and half in the other. It worked for a few years and was not ideal but done out of my friend's (admitted) desperation. In addition to the disruption to the children's education, the reason they ended up going back was that her husband was becoming limited in the number of days he could spend there for each subsequent year.
Honestly - there are loads and loads of rich people who go to the US, much richer than Harry with more to offer in terms of investment. I can't see him getting cut a deal, immigration-wise.
The LA plan, if it's permanent residence, schooling for Archie etc, means a tough life for Harry, potentially. Their behaviour so far doesn't feel like they think much of these things through too deeply.