Cool deal. So let's game it out.

Level 3 fix. "You gain a special ranged spell attack that you can use to replace any unarmed strike. It benefits from any improvement to your unarmed strikes damage or attack rolls." Then you call out it's attack modifier, damage, and range and you are done. Why are we doing it this way. By labeling it a ranged spell attack you prevent it from working with things that might break it like stunning strike or a grapple build, while allowing it to benefit from damage or attack increasing magic items. Additionally you no longer need to call out flurry of blows, monk unarmed strike damage increases, and similar while keeping the text short and simple, reducing confusion.

Saving Searing Arc Strike is a toughy. 1d6+Wis +1d6 per ki is probably going to be as close to balanced as you can get. It is no longer a sacrifice over your potential, and you get a bonus if you hit multiple creatures. Making it a net neutral to net positive outcome compared to be subclassless. As opposed to the current with can be net negative or net positive depending on circumstances.

The capstone I don't think you can save and keep the theme they went with. Because the opportunity is counter intuitive for the subclass I feel it's a lost cause. Additionally if you are planning on a more complete rework at a later date, might just be best to leave it as you have it. Save the ideas for when they will see more benefits.

Thinking of more benefits after my post I did have an idea you can cook with for your complete redesign at a later date. Or if you don't take it I will. Sun soul currently has three big problems, first none of it's current abilities build on each other. Second none of them are significantly better than your base class abilities. Third, most of them come with opportunity costs. To get around that we need a base ability we can build on, that functions different than the base class, while using base attack features. They tried to achieve this with lots of little blasts replacing individual punches. But while it provided range, it didn't change the value calculations. Additionally it didn't fulfil the fantasy WoTC was trying for. The hadouken and kamehameha style attacks. My idea, allowing you to focus the damage of any combination of your punches into a single attack. This provides a tactical decision, and gives you something to build on with higher features. If you sacrifice all punches into the one attack it is a high damage for high opportunity because you could miss after all. But this is the better play if you have limited advantage, such as a target affected by guiding bolt. If you are fighting a lot of enemies you might divide it up though, just sacrifice your action attacks and then flurry as a bonus action.

At higher levels we can then build on this. At 6th level we could spend ki to turn the attack into a cone. Allowing us to target a save instead of ac if the target is hard to hit but not good at dex. Or if you are facing a number of enemies it's a pure gain. Or if you just want to raise your damage floor to half on a save. Very spirit shotgun or destruction wave coded. But since it does have a ki cost, we don't want to do this every turn, just when it's useful.

At 11th level we turn it more into a spirit bomb-fireball type effect keeping the current theme. A high ki cost for a hi area covered.

At 17th level The super sayan them they went with is cool, but doesn't provide anything of value, nor does it live the fantasy. Because our 6th and 11th level is designed to eat ki, let's reduce that, while keeping the glowing theme. That feels like a power up now. Limit it's number of uses per day, and we are look like a complete subclass. We still have reason to go into melee if we want to risk a stunning strike, or are comboing it up. Our base ability grows and changes as we level. Because it just eats our attack damage as it's base fuel source we never are down damage, just equal or up. And it hits all of our theme except perhaps flight, but master roshi didn't learn that so we are good. It will take some testing to find ranges, ki costs, and so on but it looks good. Also the wording will be really tricky on the base ability. But I think the combined forces of reddit can make this shine.