Sound advice - "Temporary structure" is a bit of a myth. Not sure where
the term came from, but nothing in modern PP or BR really cares - at
least in a domestic setting.
To the OP:
I would say, from the POV of a "man in the street":
1) Your plot does not fit any of the ones in Planning Portal very well,
regarding placement of new erections. Mostly because you are in an
enclosed corner. Therefore making a judgement as to whether it is
permitted development or not is going to be tricky.
2) If you want to be bombproof, pay your fee to the council's planning
dept to get a certificate of lawful development or an equally valid
document of them. Someone on the phone saying "it's OK" is not good enough.
3) Or - if you are happy taking a bit of a risk: Build it.
3a) Your neighbours are happy - so they are unlikely to complain to the
council. If no one complains, the council are fairly unlikely to
investigate.
3b) If you make it past 4 years, you are pretty safe (you are not doing
a "Fidler's Castle" by deliberately concealing it).
3c) Even if the council notice, retrospective PP is a possibility;
3d) I do not think your building is "obviously taking the piss" (not a
legal term!) so I doubt you'll have a significant risk.
3e) Your biggest risk is being made to move it or dismantle it. You are
not going to be fined or risk gaol time.
Just my 2p's worth - don't base your decision entirely on it! :)