Unfortunately I cannot see any hope with the contravention itself, it's a pretty straightforward case of entering and stopping in the box when there was clearly no exit available - I can't even see any mitigating circumstances. Those two cars blocking the side street are, essentially, stationary vehicles for the purpose of the box junction rules so you would have had to wait outside the box if you had been intending to turn left also.
I'm struggling to see why a box junction has been placed there as the side road seems to be a quiet service road - hardly a prime location in need of protection but the council can place a box junction there I believe (it looks like it qualifies as a 'road').
I think (but
@Hippocrates is the expert on this) the PCN has an error in it however. From the LLA2003:
A penalty charge notice under this section must—
(a)state—
(i)the grounds on which the council or, as the case may be, Transport for London believe that the penalty charge is payable with respect to the vehicle;
(ii)the amount of the penalty charge which is payable;
(iii)that the penalty charge must be paid before the end of the period of 28 days beginning with the date of the notice;
(iv)that if the penalty charge is paid before the end of the period of 14 days beginning with the date of the notice, the amount of the penalty charge will be reduced by the specified proportion;
(v)that, if the penalty charge is not paid before the end of the 28 day period, an increased charge may be payable;
(vi)the amount of the increased charge;
(vii)the address to which payment of the penalty charge must be sent; and
(viii)that the person on whom the notice is served may be entitled to make representations under paragraph 1 of Schedule 1 to this Act;Your PCN states that they may increase the charge by 50% after 28 days beginning with the date of service of the Notice but it is required to state that this may happen after 28 days beginning with the date of the Notice itself.
Interestingly, Bexley also commit on page 2 to responding within 56 days - whereas the Act imposes no time limit. Probably worth submitting representations (particularly as they commit to reoffering the discount if you send within 14 days) and seeing what happens.