Is the "do one of the following" portion of Hwei's ability a reflexive trigger?

No, the "do the following based on the discarded card's type" portion of Hwei's ability is not a reflexive trigger.

Rule 387.1 identifies reflexive triggers by the exact phrases "Do this:" or "Do one of the following:". Hwei's text uses neither — it reads "do the following based on the discarded card's type." That's a linked instruction (Rule 359.3.e.14), not a separate trigger. The "Then" ties it all together as one sequential effect within the same triggered ability.

So when Hwei's "When I move" trigger resolves, the entire sequence — draw 1, discard 1, then check the discarded card's type for the conditional effect — all happens as one chain item with no reaction window between the discard and the conditional outcome. Opponents get their Reaction window before the whole ability resolves (when it's placed on the chain), not between its sub-steps.