This is a pretty late response but having used Aweber extensively, I can say that this method has worked well:
1) (assuming this is because of their enforced DOI) you can appeal to them directly to take off the DOI trigger (so you're all SOI) and cite an excuse about not having your subscribers sign up a second time within minutes (loses leads, revenue, etc.) - Though you should have demonstrated some trustworthiness (newsletters, etc.) beforehand so they don't laugh you out the door.
2) If you're tech savvy, set up an API between Aweber and your site (assuming you're using one) and drop your list(s) into your upload folder to have them load into Aweber (if you convinced them to toggle the mandatory DOI trigger off, you're good to go). Better to have this stagger to imitate a natural day of sign-ups (ex. 600 emails over 24 hrs as opposed to 600 emails in 1 hour)
Note, however, that 20k leads is a LOT to import all at once, especially with this method. You're best off ramping your list over a week or 2 to show 'natural' growth. Better to go slow and steady than get kicked off/banned permanently.
Hope this helps someone out there!
