Hello
I have read you job description, and if i understood it correctly, you need to transfer that website from current hosting to another one, right? What i have to ask is - do you have backend access to it (can you lo gin as administrator of that website or do you have maybe ftp access to it.
If you have any of those two, i might be able to help you. I would like to see what they did there, and backup it and test-deploy on my server. If it goes well you are good to have it for you and i would be able to transfer it to you.
If you have any questions, feel free to contact me.
Regards
Steve