Need advice as made mistake in the past

Dear all,

I have a long story to tell, hope you can stick around for a few seconds.

I've been lawfully living in the UK for the past 10 years. My tier 4 status nearing its expiry. I want to settle in the UK on ILR. But, a mistake in the past may become a catalyst for an unsuccessful ILR application. I failed to register my previous temporary accommodation with the police.

Few years back, I had to move out of my old flat to live with a friend for 6 months. It was supposed to be a temporary (call it "temp" for further reference) accommodation. Having moved to this "temp" flat, I updated my bank, HMRC, uni, council and others. Later, I moved to another accommodation which was closer to my new uni, and extended my stay on student visa few months later. The trouble is, I showed my first address, the one before the "temp" address, in the application form as my previous address. I think, at the time, I must have thought it had been too late to register my "temp" address with police. This was a wrong decision. Now, there is no record of this "temp" address with police or with home office.
I cannot remember for what reason(s) I did this. Perhaps, I forgot completely or the thought of temporary residence kept me back from registering. There was nothing wrong with my stay in the "temp" accommodation, everything was lawful and clean.

Now, I want to apply for ILR. Not possessing my old passports, I have to provide supporting documents i.e. council, hmrc, bank, doctor's etc letters to prove that I've been residing in the UK. Given that these letters will contain my "temp" address, and this address is not in UKVI records, I fear my application may be rejected.

The only thing that could possibly be useful is my details with the GP. I did not update my GP on my "temp" address and could get a letter from NHS for the whole of 10 years that contains all my registered addresses. I have had no intention to deceive.

Can anyone advice on what I could do? Not sure if I can register my "temp" address with police. It has been a few years since I moved out of that place and changed addresses few more times. All other addresses are registered with police. I do not have anything to hide. I just need to get things right.

It is what it is.
You cannot invent a new improved papertrail now.

Why not explain all this in a cogent supporting letter to support your application and thus explain the rather ambiguous, convoluted or incomplete papertrail?