I have been reading the forums for awhile but now I am probably the one who needs a little help

I have seen many cases on here but in our case the caseworker has somehow ignored the pay slips that were provided even to the extent that on the refusal letter all the rest is listed apart from the Payslips which confirms employment of my wife.
Some background:
So me and my wife have been married for almost 6 years and we live together happily. I am from Pakistan and she is from Latvia, we got married in Sept 2010 and on February 2016 we lodged our application for PR on the basis of 5 year continuous period living and working in the UK. It could have been done before but we just wanted to send and apply for PR together.
All the rest is fine apart that at the time while my wife was working and registered with WRS for the year November 2008 - November 2009 which is the start year of the 5 year continuous period she is claiming and we have provided all the necessary documents P60's and payslips (the payslips are covering the whole year period but I sent them 4 over different months beginning Nov 2008 then two in the middle and last one for November again just to keep the burden less my fault maybe) P45's letter of employment where available etc.
To cut it short the case worker has either overlooked or somehow missed the payslips completely and has focused more on the p60 which runs from April to April so he has concluded that not enough evidence has been provided to cover that whole period btw November 08 to November 09. I can assure we did provide payslips and they have been sent back as well but there is no mention of payslips any where on the letter that either they were missing or are not sufficient.
Although it has been refused but Appeal is granted and there is no requirements to leave the UK as such at this point. So I wanted to ask shall I send in paper work again to case worker with a cover letter for reconsideration or make a new application or go via appeal. My own understanding is to write a cover letter carefully and send it all for reconsideration. I can tell you from the letter we have received that there are other little mistakes as well on the letter of refusal which shows negligence or caseworkers incompetence or lets just say errors.
Also to include we can also apply on the basis when Latvia came out of accession state 01 May 2011 - 01 May 2016 provided we have supportive documents that she is or has been a qualified person, which again as I said we can but in this case we would need to make a fresh application as well.
I have never had any issues before with Home office as I always send them a cover letter explaining everything in details but this is the first time ever I didn't and thought it should be fine which was certainly not the case.
I would greatly appreciate some feedback from the honourable people of the forum.