Why do you care about the snails? Pond snails are largely harmless.
I'd also suggest assassin snails. They're unlikely to go after your big snail, and, yes, they breed pretty slowly. You start with 5, you'll have maybe 10 in a year. They burrow, also, you don't see them much, and they're kinda pretty.
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My current project, a 65 gallon aquarium stocked with vernal pool fauna.