But, printing $i still seems to hold the large number despite $i not being a Math::BigInt.

I'm betting $i is just a string. useDevel::Peek; Dump($i); would confirm that.

Two pedantic notes,

• Your do { ... }; blocks could have been written as { ... }. It would make a difference if you used next, redo or last in the block, however.
• I don't see anything that says that bignum is lexically scoped. It's misleading to place it in a block if it's not.

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