Solar Brawl, as you note, is meant to synergize with Athletics as well as Resistance. Those three especially. There's a lot of counters to the most degenerate stuff Solar Brawl can do. Run away, strike from range, hit them REALLY REALLY HARD. Athletics and Resistance both contain means of mitigating these counters.
The Attack From Base Initiative Engine
For this one, you want to dip Athletics. Athletics has Increasing Strength Exercise. This adds to your Strength directly for 3m or 3i per dot, one dot per point of Essence you have. At Essence 1, this is Strength +1, which is +1 withering damage, yes, but also +1 raw decisive damage because that's how ISE rolls.
You want to be a Dawn. The Dawn Anima adds +Essence/2, round up, to your initiative. +1, +2, +3 at E1, E3, and E5 respectively. So, now, with an Essence 1 Solar, we have +1 raw decisive damage, +1 base initiative. When we attack, we reset to 4i, and add +1 for 5 base damage on any decisive attacks that we make. This beats Light Artifact hardness, which is 4, but loses to Medium, which is 7. Hold onto that, though!
Enter One With Violence and Striving Aftershock Method. They have Falling Hammer Strike as the core prerequisite to purchase them so they're integrated with the Onslaught Engine. One With Violence awards the lesser of (Essence or 5) as a bonus to the initiative gained from Crashing an opponent, as a free, permanent effect. Initiative break is permanently enhanced, growing in power as you do, capping at +10 for crashing someone at Essence 5. Striving Aftershock Method lets you spend 2m to gain +2 Base Initiative upon resetting after a decisive attack.
Both are Essence 2 effects, young Solars get them, though baby Solars need Supernal. So, we're up to raw decisive damage of 7 now for an Essence 1 Solar with the Dawn Anima, ISE, and Striving Aftershock when attacking from base initiative after making an initial decisive attack. We now can also freely attack opponents in medium artifact armor and many forms of megafauna.
Adamant Fists of Battle. This is a very important scenelong Charm. Add Strength to minimum damage, so +5 because you're a Solar Brawler, and double 10s on decisive attacks. This is an Essence 3 Charm, you need Supernal, Ferocious Jab, and Burning Fist Burial to qualify. Our base initiative attack is now 7 dice, double 10s on the damage roll, defeating medium artifact armor unless someone has one of the extremely rare "increase armored hardness" effects.
Increasing Strength Exercise, Thunderclap Rush Attack, Falling Hammer Strike, Ferocious Jab, Burning Fist Burial, Adamant Fists of Battlex2, One With Violence, Striving Aftershock Method. We're up to 9/15 Charms. for an E1 Solar...but 9/20 for an E2 Solar, and Adamant Fists as our only Supernal pick.
You probably want Wind-and-Stones Defense and maybe Iron Battle Focus for your basic defense options here, but it's not mandatory to pull this trick off. Call it 11 out of 15 anyways.
So the next one up is fun! Orichalcum Fists of Battle. Flashy, expensive Charm. 8m,3a, 1m per turn of upkeep, but we'll get back to that. You can generate the anima easily by maxxing JB in order to make an immediate Decisive attack with Adamantine Fists of Battle out of Join Battle. This lets you turn Adamantine Fists into a supplemental Charm so long as you damage them, saving you a precious turn of set-up. Unless you get a bizarre JB roll you'll almost certainly have more than 10 decisive dice, so just attack whoever you're certain you can hit for 6m,1wp to set up Adamantine Fists as a Scenelong Charm which enhances this attack, plus 10m on Join Battle to bring you to +2 Anima from JB, +1 from your actual attack.
I'll note, this can be a Withering attack, and it should be a withering attack if your opponent has any anti-decisive panic buttons or seems like they might that could threaten your ability to activate the scenelong effect. A decisive strike from a Solar Brawler must be avoided if at all possible because of how Heaven Thunder Hammer is triggered, it's a post-roll activation based on damage that does a huge chunk in one big burst. It's totally opportunistic, and they need to deny you the opportunity.
What you're actually doing is so so much worse, of course, but they don't know that. A Withering attack with no accuracy boost is an attempt to pressure you into spending motes. Take it on the chin, tbh, try to avoid with your defense, do the Shounen thing of exchanging basic hits while figuring each other's combat style out. A decisive is a set-up to HTH and must be evaded at all costs. Fun bit of tactical psychology at play!
So, Orichalcum Fists, though. Once you have Adamantine Fists up as a scenelong effect, next turn you activate Orichalcum Fists. You now ignore Hardness entirely for the scene and reroll 1s and and explode 10s on the damage roll for decisive attacks. It also increases base initiative by 1.
At Essence 1, we now reset to Initiative 3, +1 Dawn Anima, +2 Striving Aftershock, +1 Orichalcum Fists, +1 raw damage from ISE for a total of 8 damage dice. You ignore hardness, so you can't ping off people. You double 10s on the damage roll, you roll 1s away so you have more chances to roll 10s, and 10s explode, you roll them until no more 10s show up, keeping each result of 10.
12/15 Charms. Congratulations! You no longer need to make Withering attacks! You roll something like 5 health levels of damage per hit on average. 1m per turn upkeep, but just have a high accuracy pool! Strength 5, Dex 5, Stamina 5 is hard to get at chargen but you can do Strength 5, Dex 5, Stamina 3 and raise that over time.
For the last Charms, I strong recommend that one purchases two Ox-Bodies and a Charm of your choice for maximum hilarity. If you're using Advanced Chargen, Spirit Strengthens the Skin, Durability of Oak, Iron Skin Concentration, Ox-Bodyx2. Grow (Stamina) -0 health levels the first time someone seriously hits you with a Decisive and have 12 health levels before becoming incapacitated.
And this is just Supernal Brawl and E1 Resistance! Resistance further Durability-of-Oak Technique, 6m, halve all Withering damage for the whole turn, Essence-Gathering Temper, regen motes based on soaking damage, Adamant Skin, soak decisive attacks, Battle-Fury Focus, cut your wound penalties and gain +1 bonus die on attacks to save on your Excellency, Bloodthirsty Sword-Dancer Spirit, cut all wound penalties, gain another +2 bonus dice to attacks, and regen a mote extra per turn to spend on combat effects, canceling out the upkeep or Orichalcum Fists.
Athletics has Monkey-Leap, Graceful Crane, Spider-Foot, Racing Hair, Mountain-Crossing Leap for escaping and running other people down, to say nothing of Thunderbolt Attack Prana (aka "kaio-what-?!").
I might come talk about how fucked up grapples are later if anyone is interested because Solar Grapple is so fucked up.