Update: after my last reply to GTR informing them that I’ve itemized everything and happy to share my spreadsheet if it would help, I received a reply back the following day with a settlement figure and their own spreadsheet which itemized all the short fares. Our figures were very similar (theirs was around £100 more) so I agreed to the payment which was just over £2.2k. That included £300 for their admin fees.
They used the difference of what I had already paid and what I should have paid to calculate the difference.
To anyone in a similar position, I recommend in the first instance that you deal with GTR directly without a solicitor. Just be genuine, honest and don’t try to hide anything from them.