Sadly, I'm not sure your player can make it work.
In the large majority of cases, DnD combat is balanced under the assumption that players will be dealing significantly more damage than they take. A typical encounter sees PCs fight a group of enemies that have roughly the same amount of HP as the players, if not much more, and players typically win an individual successful fight without having come close to running out of their own HP. 5e is even built around the idea that no individual combat should pose a serious risk, even if its challenge rating is relatively high, and players primarily risk death by attrition over the course of multiple battles before resting.
As a result, even if the curse your player took meant that they only took damage equivalent to what they dealt, they would be unable to contribute much to combat. A single point of damage against an enemy is less valuable than a single point of damage against a PC, so every time your cleric tries attacking, they end up expending more of their party's resources than they gain. The most optimal strategy is to just stay out of the way.
The fact that the cleric takes three times as much damage as he deals is so extreme that any time the cleric attacks, the result overwhelmingly sets the entire party back. Either the cleric deals negligible damage in exchange for becoming quite a bit more vulnerable, or hampers their entire team by pushing themselves close to death for damage that any other player could deal with absolutely no negative consequences whatsoever. When the cleric becomes weak, the party is going to have to adjust its strategy to protect him, which will affect the flow of combat for the entire group.
The cleric can use other strategies, but they will only get your player so far. Spells and items are expendable, so they can only be used a handful of times between rests. And support cantrips do give your player other ways to influence battle, but they are all both situational and weak, so the cleric will rarely get a chance to impact a battle with them. As a result of these limitations, no character can plausibly use support techniques on even half of their turns. (In the campaign I'm in, I'm playing a very support-focused bard, and even though I love hanging back and controlling the tide of battle, the large majority of my turns are spent firing a crossbow.)
The balance of DnD 5e means that the most common way for players to engage in combat is by dealing straight damage. And, unfortunately, his curse is so extreme that he has locked his character out of that.
You said that your player chose this curse and has more DnD experience than you. I would suggest talking to him directly about why he chose this curse and how he expects to be able to contribute in battle in spite of such a large handicap. He might have some specific ideas, which you can then plan around. Or he might realize that the curse is more of an impediment than he expected, in which case you can retcon it, weaken it, or give him a way to remove or mitigate it.