We prohibit automated entries though and each mail-in entry needs to be handwritten sent in its own envelope. The postage costs more than the EV of one entry, so we don't expect this to be an issue.
From my research, it sounds like, because prize linked savings accounts are so new in the US, it's unclear whether requiring someone to have an account would count as consideration. By offering an AMOE are you taking a conservative position on this until it's more clear? Or, has a judge/court ruled on this?
I was involved in a sweepstakes at a bank a couple years ago and we took a similar route as you, offering a mail in AMOE, requiring handwritten entries. Without any marketing, we received several thousand entries, mostly from people who mail in entries to every sweepstakes they find.