How do you limit ticket sales made on the day, bearing in mind they will be being made at stations, TVMs▸ and on trains up and down the country?
For the Champions League final in Cardiff earlier in the year
GWR▸ made all Cardiff bound passengers require a reservation. Sure it was a pain, but they did it. And no reason why they couldn't have done that here too (hell it would have probably been easier than Cardiff due to the nature of it being a branch line).
I am not sure if that would actually help or not, but it certainly isn't impossible as you seem to suggest.