To me, it doesn't really sound like a capacity issue but more of an issue where they need to validate individual tickets?
If it's a matter of validations, surely this can be a batch job that can be done? Surely with today's computing power, it can be chewed though with relatively ease.
Lastly, if it's 3-4x their usual activity - surely that if in the past, results can be checked at 10pm or beyond - why is it that so many hours now since the draw - you still can't log on?
If it's capacity issues, why can't they just scale it using a service like AWS?
Even if their database is housed in their HQ, surely the website could be scaled and elastic whereas the endpoint could be throttled via an API to their in-house 'database'.