score:5
Officialy, most likely not.
Applying to the Home Office yourself will probably lead to a failure.
You must employ a well chosen Immigration lawyer, that will collect your previous history (abused child before entry into the United Kingdom), which lead you into entering under deceitful circumstances (providing a US Passport Card - which was probably assumed to be a European ID - in Northern Ireland instead of your Passport which contained a refusal of entry stamp).
What you have done since your entry:
in short, in the last 3 years have proven yourself a (possibly) useful member of society.
A further note: this process should be started before any official notice has been made that you are illegally in the UK (i.e. you are doing this on your own initiative to rectify the situation).
A good immigration lawyer will know which organization / forum this information should be placed, in a convincing manner, so that they would 'suggest' to the Home Office what proper decision should be made.
As far as I can tell, based on your previous questions and resulting comments, this is the only viable option you have to achieve your objective.