You might be reading this too late but I would go online now and see if there are any advance tickets on offer either between Euston and Carlisle or Kings Cross and Newcastle; they may work out to be your cheaper option without a railcard.
If I were you I would order a new railcard through ATOC because you can get this for a cheaper cost with a crime reference number. They would also deliver it to any address you give at the time of reordering so it could be sent up to Carlisle for you to use on your return journey (i.e. if you were travelling back on say an advance single).
If you do end up traveling on a full fare ticket, I would keep hold of the tickets, your crime reference number (and any paperwork) along with the replacement railcard. You could send this (of course not the actual railcard but a copy of it and the receipt as this will show it was a replacement one for one that was stolen) into Virgin (either East or West Coast whoever you end up travelling with) and ask them if as a goodwill gesture if, given you were the victim of a crime that has been reported to the police, they would be willing and kind enough to refund the difference between the railcard and non railcard fare. I have found Virgin West Coast to be very reasonable and helpful when there have been legit circumstances. They may be more willing to do that too if you say you would be happy to accept Rail Travel Vouchers (hint!)
If you are "lucky" enough that one (or both) of your services are delayed by 30 minutes or more then you would get some of the money back through delay repay too - yes in RTVs but you can use those to add onto your Oyster card too so could use it right away. Also, if time is not of the essence how about - as a one off - you see what fares you can get on Megabus (and National Express - not sure if they service Carlisle though).
Hope this helps you find something of an affordable fare!
