Well, that claim that the Whitehouse told the UK can't allow Ukraine to use Stormshadows in Russia proper is a bit confusing.
The U.S. has absolutely no jurisdiction over the UK's policies AND just yesterday, Ukraine rained U.S. HIMARS down on a Russian training facility south-east of the Kursk region.
There is also this interview with State Department Matthew Miller on the subject:
State Department spokesperson Matthew Miller explains the U.S. position on Ukraine's use of weapons in an interview with VOA’s Ukrainian Service