Did the Byzantine iconoclasts have better relations with Muslims than iconodules due to a shared animosity towards religious imagery?

In short, no, the policy of Iconoclasm had very little if any effect on the relationship between the Caliphs and the Basilius’s.

I should preface this by saying that Iconoclasm is probably the most difficult period of Byzantine History to really write about. our primary sources, Theophanaes the Confessor and Nikephoros, were present at the councils that overturned Iconoclasm and were heavily Iconodule. Their heavily Iconodule leanings basically means that everything they write is suspect or just straight up completely useless. Modern scholarship has begun to fundamentally reassess the period of Iconoclasm, and it’s now believed that it was not nearly as big of an issue for the Iconoclast Emperors as it is commonly believed to be. For example, at the 2nd Council of Nicaea, where Iconoclasm was undone by Irene, it was one of the last topics to get addressed, something you wouldn’t expect if the issue was tearing the Empire apart like Theophanes suggests. Addressing the influences and inspirations for Iconoclasm is worthy of a question on its own, which if you were to ask I would happily answer : )

In the first period of Iconoclasm, I really find it hard to believe that there was really ANY difference in relationships between the two powers. Leo III, who supposedly was the first great Iconoclast (which there’s really very little evidence to suggest was actually the case), was indeed the man who beat the great Arab siege of Constantinople. Following this, Leo continually had to beat back large Arab raids and fought major battles such as Akroinon during his reign against the Arabs.

Now, your question is something that I’ve heard discussed before by people referring to Constantine V, but even he I would argue really had no difference in relations with the Arabs for any reason due to Iconoclasm. Constantine V’s reign did indeed see a relative breathing period of Arab raids, but that was due to outside factors. For one, his armies were simply victorious in battle, so it’s harder to raid a territory when their Emperor is battle tested and whose army has crushed yours multiple times. Alongside this, Constantine just so happened to be the ruler during the Caliphate falling apart under the reign of Marwan II and the overthrow of his Umayyad dynasty. Arab swords were simply too busy slaughtering each other to be used against the Byzantines, and so that’s why you see a relatively peaceful period (although this did not stop Constantine from crossing the Tauros mountains and campaigning in Cilicia and Syria). If anything, Constantine used the bad relations (e.g. him killing lots and lots of Arabs), to strengthen the legitimacy of Iconoclasm, as it seemed to indicate divine approval of the plans.

Arab raids would continue on throughout the reign of Constantine’s successors, including his son, grandson, and future daughter in law, and these large scale raids wouldn’t really stop until well into the Macedonian period when the caliphate fractured and the army was just so overwhelmingly powerful that Arabs knew that anything other than local raids would just result in their heads on spikes.