If my opponent uses Hard Bargain, then I counter with Lilting Lullaby, then can my opponent still play an En Garde?

Yes, your opponent can play En Garde in this situation.

The Chain State

1. Your opponent plays Hard Bargain ([Reaction] spell). It is on the chain.
2. You play Lilting Lullaby ([Reaction] spell) in response. It is placed on top of Hard Bargain.
3. The chain is currently in a Closed State.

Why the Opponent Can Play En Garde

1. Reaction Timing: En Garde has the [Reaction] keyword. Per Rule 813.1.c.1 and the [Reaction] keyword definition, cards with [Reaction] can be played during Closed States on any player's turn.
2. Chain Resolution: Because the chain has not yet finished resolving, players may continue to add more [Reaction] items to the top of the chain (LIFO - Last-In, First-Out).
3. Lilting Lullaby's Restriction: The restriction imposed by Lilting Lullaby only takes effect once that spell finishes resolving. It does not prevent the opponent from playing a [Reaction] spell in response to Lilting Lullaby while it is still on the chain.

The Sequence

1. Hard Bargain is on the chain.
2. Lilting Lullaby is on the chain.
3. Your opponent plays En Garde as a [Reaction] in response to your Lilting Lullaby.
4. The chain consists of:
   • Top: En Garde
   • Middle: Lilting Lullaby
   • Bottom: Hard Bargain
5. The chain resolves in reverse order:
   • En Garde resolves first.
   • Lilting Lullaby resolves next, countering Hard Bargain. (Because Hard Bargain is countered, it is cleared from the chain and does nothing per Rule 425.1.a).
   • The "can't play spells" restriction applies to the controller of Hard Bargain only after Lilting Lullaby finishes resolving.

Note: I could not fully verify this ruling against the retrieved rules.