The Islamic Republic has always relied on terrorist organizations to do its bidding. Since the 1980s, Tehran’s most operationally savvy protégé, Hezbollah, has given the regime the ability to manipulate Lebanese politics and kill scores of its enemies, including U.S. troops. In the aftermath of 9/11, especially after the 2010 Arab Spring, the Iranians fine-tuned their grand strategy. The collapsing Arab state system allowed the mullahs to assemble nonstate paramilitary outfits that they could deploy to various battlegrounds. Iran-aided militias helped evict the U.S. from Iraq, ensured the survival of Bashar al-Assad’s dictatorship in Syria, and mauled Saudi Arabia and the United Arab Emirates in Yemen.

The axis of resistance has transformed the Islamic Republic into the region’s most essential power broker. No government can form in Iraq, Syria or Lebanon without its consent. Iran’s auxiliary forces also deter its enemies. Should Israeli or American leaders consider striking Iran’s nuclear installations, they have to take into account Hezbollah’s formidable arsenal of missiles. And the war in Gaza, for which the Islamic Republic had long been prepping Hamas, has reminded Arab potentates that expanding the Abraham Accords carries enormous risks.

Yet the axis of resistance needs Hamas to survive in some form. Going full martyr—Sunnis die, Shiites watch—would leave Iran caged and embarrassed. Iran’s theocracy has signaled its intent. On his recent tour of the Middle East, Iranian Foreign Minister Hussein Amir-Abdollahian warned: “If the Zionist aggression does not stop, the hands of all parties in the region are on the trigger.” Nor was he being coy about one set of hands. “The whole world knows that Seyyed Hassan Nasrallah”—Hezbollah’s leader—“is a man of action and has played an outstanding role in securing the region and Lebanon.” There have already been modest clashes on Israel’s northern border.

Behind Iran’s incremental war strategy is a reasoned diplomatic one. Supreme Leader Ayatollah Ali Khamenei hopes that a gradual expansion of the war might hasten its end. Arabs will protest more. Europeans will dispatch mediation missions. The Biden team, which seems terrified of a larger conflagration, is already encouraging patience in Jerusalem. Washington has been arguing for limited military intrusions that would kill fewer civilians. Anxious Israeli generals, who would need to keep larger ground and air-force reserves for a more active northern front, may start talking again of “mowing the lawn” via periodic small incursions in Gaza, even though that tactic has failed miserably. Mr. Khamenei knows his regime can restore a degraded Hamas. If Hamas loses too much manpower and leadership, however, it might stay dead even if the spirit of resistance lives on.

The trickiest question for Mr. Khamenei is how an escalated conflict would affect the regime’s standing inside Iran. Direct retaliation by Israel, and especially by the U.S., might trigger a chain reaction of discontent with little rallying-around-the-flag effect. The Islamist regime is wobbly. A struggling economy and a rebellious public scornful of Arab and imperialist Islamist causes—Iran’s “forever wars”—weigh on the supreme leader’s decisions. The more direct the U.S. and Israeli threat is to the regime, the more likely that Mr. Khamenei will retreat. An explicit American threat to take the war to Iran would give Israel more breathing room to dismantle Hamas in Gaza, if that’s what Jerusalem decides to do. President Biden’s decision to bring two aircraft-carrier groups into the region helps. He should go further.

What the White House shouldn’t do is quietly warn Tehran not to meddle in Gaza or to unleash Hezbollah. The Islamic Republic is accustomed to back-channel admonishments. America’s armadas have patrolled the Gulf for years without sufficiently changing the mullahs’ calculus. To make a lasting impression on Mr. Khamenei, Mr. Biden needs to declare publicly a red line: Another Hezbollah missile attack on Israel will invite direct U.S. retaliation on Iran. In 2003, when Mr. Khamenei feared the possibility of the Bush administration unleashing its “shock and awe” warfare on Iran, the clerical regime suspended its uranium enrichment. When the perennially unpredictable Donald Trump killed the Islamic Republic’s famed commander Qasem Soleimani, Mr. Khamenei let loose a short missile barrage at U.S. forces in Iraq but went no further.

Iranian escalation this time around is a certainty. Jerusalem and Washington need to deny themselves wiggle room and threaten the clerical regime, not its proxies. This war is going to get worse. It’s past time for Israel and the U.S. to up the ante.

Mr. Gerecht, a former Iranian-targets officer in the CIA, is a resident scholar at the Foundation for Defense of Democracies. Mr. Takeyh is a senior fellow at the Council on Foreign Relations.