I too have noticed that on the occasions when I do have an early morning poo it is often a 5 or a 6, whereas later on it’ll be a 3 or 4, or maybe even a 2. Your observation has made me feel that working with what happens naturally probably really is the best way, if it can be achieved.
Although I suffer seriously from shy bowel, my pelvic floor had become weak and I was in significant danger of not being able to keep a poo in long enough to get to a toilet, once the urge came. So I too became fixated on having to have a poo in the morning before going out.
I tried getting up absurdly early, eating something as soon as I got up, and going for a gentle early morning walk – all to get things moving. I had varying degrees of success, but certainly nothing I could really rely on. I even tried a glycerine suppository once – once was enough – to get the job out of the way to free me up for the day.
I came to the conclusion that trying to force an early morning poo just wasn’t going to work with any regularity for me.
For me a “natural” time is about 2.5 hours after I’ve got up – and only if I’m not under any time pressure. Otherwise a “natural” time could be any time – late morning, afternoon, evening or even the next day. Missing a day makes me feel a bit yucky, but not as bad as you describe.
I began to view myself as semi-housebound – I didn’t dare go out till after I’d done it. Holidays in groups became a no-no. I even wondered half-seriously whether a voluntary colostomy would be better – at least I’d be able to go out.
Fortunately doing pelvic floor exercises religiously has helped enormously with the urgency problem so once again I dare go out pre-poo. I have had excellent help from the NHS incontinence service for which I will be eternally grateful. I self-referred myself to them having seen an advert saying you could in the loos of a motorway service station, thanks to the Bladder & Bowel Foundation. (They haven't been able to help with my shy bowel problem - just the urgency/weak pelvic floor.)
I have incidentally given up caffeine too – but because of urinary problems, not bowel problems.