The webmail is one thing I could try. I have been transferring files use Google Drive and emailing the link. I was looking at other email clients but I could not find any that would import Mail accounts and emails. The real problem is apps that create email like QB and LR etc. will create an email but attachment aren't in the email.Do you have a webmail interface to your account? Doesn't fix your problem with Mail but it might let you get your file(s) out.
FWIW, I use Thunderbird as my default mail client. I keep Mail open for Spotlight indexing and for the occasional time when I want to create an email from a file I want to send as an attachment (drag file to Mail icon).