First, it's explicit in the Constitution that "Each House may determine the rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two thirds, expel a member." Whatever rules a House likes for its proceedings are the rules, and whatever punishment it designates for violating them is the punishment. The case law on that goes on to state that this means that the courts may not hear a case on such matters; no Federal court has the authority to even hear a case on the rules, much less get to the point that it can rule whether something is free speech or not.
Second, the Speech or Debate Clause only protects members form being held responsible "in any other Place"; their own House is perfectly allowed to hold them responsible for what they say. In accordance with the previous bit.
Third, this isn't a law, it's a proposed rule of the House, in the decidedly non-public forum of the floor of the House. The First Amendment doesn't remotely apply, at all, either literally or in any of its court-extended meanings. Even if the courts were allowed to rule on the rule (see the first problem), current precedent would fall on the side of the rulemakers.