The annual rainfall isn't so much of an issue as how much you get in one go (before your water butt can drain). One millimetre of rain works out to one litre per square metre so 23 square metres of roof means you can collect 26mm of rainfall. I'm south west of you, but we probably get that in a few hours a few times a year.
Here's a calculator that should help you. Plugging your numbers and assuming the roughness value they prefill means you can drain 13 litres per minute (if that's 20mm bore, I'm not familiar with MDPE). That sounds to me like it would work even with some additional losses. But you would want to maximise the head, and consider what happens to any water that overflows, as this will be when the ground is already wet.
The soaker hose does need a bit of pressure, and it would dump the water onto wet ground - the advantage over just enough gutter to keep the rain away from the door probably isn't significant.
I would do it slightly differently - I'd take the pipe out near but not right at the top of the water butt. This would decrease the size of the buffer but increase the head (in a ground level pipe) and therefore the flow rate, while giving you some stored water to use in summer
