Should I read into those phrases the implication that you think I am making things harder for myself than need be? I hope that is the case and you can tell me a simpler method.
I think the way in which you have shown grim determination to tackle what most realise is a highly complicated and some times inexplicable set of rules, regulations, instructions, and forms, and not run away to hide under the bed (as many do) is admirable.
Many of those approaching the US Tax Code for the first time are surprised to learn that, in many instances, there is no 'one' correct method of calculating a tax return. In many instances there are multiple ways. None are the sole correct way. In addition, there are many situations in which there are only opinions or interpretations of 'a' correct way to handle a given set of circumstances. International tax advisors may hire a tax attorney for a fairly substantial sum simply to review a set of circumstances and give an interpretation on which that tax advisor feels comfortable following. At times, The Code is that difficult to comprehend.
Someone said just use the dates you received income and dates you paid tax.
If I remember the conversation correctly, that was Guya. Guya is a highly (internationally) respected tax advisor specialising in US/UK tax. Even though I am a pure know nothing average punter, I may at times disagree with him on some topics whilst fully understanding his stance; but I will always respect his opinion. You may be wise to do the same.
The US/UK tax situation is additionally complicated by the use of two different tax years. The US has 1 Jan. to 31 Dec., the UK 6 Apr. to 5 Apr. of the following year. You know this very well, and the resulting difficulties that can arise.
Form 1116 has two options: the accrual method and the cash method. If I remember correctly, the cash method may also be used as an accrual method (a happy coincidence for the US/UK situation). I may be wrong about that. Your situation, again if I remember correctly, has three income sources: pay, interest, and qualified dividends. Your struggle with the dividends reminded me why I only have bank/building society current and savings accounts and nothing else. It may seem unfair to rule ones self out of a number of investment opportunities, but that's the downside of being a US Person unless you have the funds for employing an advisor, or a rabid interest to fully research the US Tax Code (and there are some on this site).
Most average individuals utilise the cash method for form 1116. If you have a very complicated tax situation; or you are a HNWI with angles to play; or you are a large international company; or you have lived in a country like France as a US Person; you will use the accrual method.
I view your situation as being a fairly basic situation (pay and interest) with the additional problem of having qualified dividends. For the pay and interest, the simple solution IS to declare all income received in the US tax year, and declare all taxes paid ONLY in that same US tax year, regardless of the source of the income. There will be years in which you come out ahead, and years you come out behind, but over time it WILL even out. I do not envy you the handling of the qualified dividends, but I'm sure a professional would advise a fairly straight forward way of handling that.
Finally, I would make the following suggestion:
The IRS can not fault you for making every effort to complete your tax return correctly. One of the areas I may disagree with Guya on is this: make your best effort to be correct with your understanding of the rules; note why you chose to do as you've done in the completion of the form; and then submit the form. If the IRS disagrees with you, they will let you know. You may hear nothing more, or they may disagree and you may suffer additional taxes and an interest penalty on a late payment; BUT you will now know the correct way (or the way the IRS deems the correct way, which isn't always correct), and you will know how to prepare your return in the future. You've paid a price, but you've gained knowledge.
Again, all of this is only my 'lacking of knowledge' opinion. The important thing is for you to submit a tax return in which you feel the way you have prepared it is the correct way. That's all you can do. Don't over-stress. If it's wrong, you'll be given an opportunity to correct it. If you've made a sincere effort, you are not a tax evader nor a criminal and the black helicopters are not going to land on your roof; you're just the average tax payer with foreign sources of income.