If not an 8833 then what documents do I provide to the IRS or HMRC to demonstrate where my centre of vital interests belong.
You may be OK, but my honest opinion would be to talk to a pro. This isn't to find someone to do the tax returns, but to obtain an opinion about any problems you
may have given your circumstances. Perhaps a 1 or 2 hour sit down to talk through alternatives. I'd rule out a pro in the US unless you know for sure they're competent in US/UK tax. Obvious suspects would be those in the UK.
Start with the UK. I take it you won't arrive until after 06 April 2016. That would put you into the 2016/17 tax year. The drop dead last date for filing a 16/17 UK return would be 31 January 2018, so you have plenty time to work out the UK side (and to have sufficient days in the UK by then).
For the US, the 2016 return is 15 (April(?), June(?), October(?)) 2017, all depending if the US recognises your residence abroad on 15 April or if you were to file for an extension. Still, there's time to obtain an opinion and act on it.
Both the IRS and HMRC will be happy to receive a return, but as you've obviously figured out, the trick will be to have one recognise a tax residence in the other country in order to have treaty protection. They'll both be happy to receive a full amount of tax due, but you won't be happy with that scenario, so your goal is to establish a residence that will hold up to questioning. If you're living in the UK for 8 months out of the tax year, they'll want tax for that year and no treaty would prevent it, only possibly the offset of credits IF they accept them (back to the 4 point test).
As an American living abroad, I know I'm extremely cynical about US intentions, but the one objective is to make sure the IRS doesn't view your UK residence as a 2nd, or vacation home. That would apply whether you're renting or an owner. Think about it, their view might be you're claiming permanent residence abroad simply to avoid paying US tax via a treaty with the UK. Yes, the UK has higher rates, but that's not been a problem to prevent them jumping to conclusions up till now. Welcome to the world of the US citizen abroad automatically guilty of avoiding paying their fair share of US tax. Sorry to be a bore on this but that's the direction things are going at the present.
Filing an 8833 with the 'centre of vital interests' may achieve no more than a large red flag on your return. Right now, I'm not sure the computers are linked that would record your entry/exit to the US for the IRS. Therefore, how you file (as if you have moved permanently, or not) will start with your return. Here's where professional advice would help a great deal.
You know you're doing nothing wrong, simply want the joy of being able to be in both countries, and you have the citizenship entitlement to do that. There are others on BritExpats we both know of who are doing the same thing, but as I understand it, after setting up a home in the UK, he is carefully limiting his time in the US to 2, two month visits per year (or 122 days per year) so he falls below the substantial presence test. It sounds as though eventually you'll be doing the same. Unfortunately, tax authorities have suspicious minds.