I think my original question was flawed. I've been reading through the HRMC documentation (again :-)), and now I'm thinking split year taxation only applies in cases where you pass the Statutory Residence Test (SRT). For tax year 2021-22, under the SRT system, I have no automatic inclusions/exclusions, and just the one UK tie. If I'm reading it right, that means I can spend up to 182 days in the UK that tax year and still be considered non-resident. i.e. as long as I spend less than 183 days in the UK, I won't be taxed on my US gains until the following (UK) tax year.
I'm only about 80% confident of this interpretation as I half remember something overriding this.
@tubaleiterGood points, but my bad, I should have made it clearer I'm originally from the UK (moved to the US 18 years ago when my job moved here) and have UK citizenship (so no visa required). I have a UK bank account and rent out my old UK home (which I have to file UK tax forms each year).
I did once have to use the NHS while non-resident. In theory, as a non-resident UK citizen it's only free if you intend to be staying in the country for 3 months or more (or, at that time, were resident in a EU or other country with a reciprocal agreement). I was honest when I went in for treatment and tried to pay (it would have been covered by my US insurance anyway), but the staff had no idea how to process it and basically just took contact details and I never heard anything from them ever again.