Does it need to be an invoice? A letter from the hospital explaining the teatment and cost might be sufficient. UK banks are now very proactive in trying to protect customers from scams due to the increasing number of cases. Significantly, these often result in poor publicity attaching to the banks when they refuse to compensate for their customer's stupidity.
I use Wise to make transfers from the UK and agree with other posters that they're quick, efficient and cheaper than UK banks. But those are transfers to accounts in my name, there may well be similar questions and reticence from Wise if sending a substantial sum to someone elses bank account.