First, I assume the problem isn't as much 1M files, but your server unable to handle it? Logically it worked elsewhere, right?
My opinion is that presenting is as a “problem” to the client might be a little one sided.
It's in realm of technical possibility, but it won't be a smooth process. WP stores image sizes information as serialized PHP data in database. So for each image it would be quite involved operation of adjusting that data. I don't think I've seen anything that automates it away completely for such use case.
My strong concern also would be URLs changing. I imagine site with such quantity of images is likely to receive not insignificant amount of traffic from image searches. Messing that up would be very unfavorable.
Overall client should be probably presented with upsides and downsides of the change, more so if you are going to be charging for it. The best thing for them (and maybe for you) might be to look for system that better accommodates their needs.