Perhaps what has happened is that although Advance tickets are not refundable, the customer service agent saw a booking for August being asked to be refunded "well in advance" in June and applied discretion, but later realised that the booking had been changed "last-minute" just 3 weeks before the original travel date, and they would not have applied discretion to refund that booking on such short notice? (Sort of like the old trick where if you're past the cancellation deadline before a booking at a hotel or fancy deposit-taking restaurant, you instead change the booking "last-minute" to much later and then cancel the new booking "well in advance" of the new cancellation deadline for a full refund...)
Anyway, I'm still a bit confused about what happened to the tickets...
According to this message, you bought ticket 1 for August, changed ticket 1 to June, bought ticket 2 for August, then paid to move one of the tickets to August. This cannot be correct since you would have been left with two tickets for August and none for June to travel on.
According to this mesage, you bought ticket 1 for August, paid to change ticket 1 to June, then bought ticket 2 for June. This cannot be correct, since you would have been left with two tickets for June and none for August to refund.
What exact purchases and changes did you make, and to which ticket? It seems that at the very least you made one change from August to June (to be able to "go earlier"), and one change from June to August (to be left with the ticket whose refund is being disputed).
If you changed the ticket from June to August at the same time as you bought a new ticket for (presumably a different time in) June, then perhaps you could beg LNER to reinstate the refund based on the fact that you could equally, for the same cost and with the same resulting tickets, have changed the June ticket to June and bought a fresh change-history-free one for August which would have then been refunded under their discretion - but you would need to give an exceedingly clear and concise explanation of what happened, and the request would need to be very polite as it ultimately hinges on a discretionary refund of a non-refundable advance ticket.