[] Send Ranca. Ranca already wanted to duel the Viscountess, you can argue that she has precedence there and have her duel in your stead. She will probably win but if she loses? Again, that would make your accusations against the Viscountess be generally viewed in society as false and anything she does in retaliation 'putting you in your place'. In comparison to personally duelling this would mute the advantages of winning as well as being a lot more likely to succeed.
"I accept your challenge." You answer across the open channel. "Baroness Ranca, I believe you already stated your wish to face the Viscountess and so have precedence, would you fight in my stead?"
There is no hesitation, not that you expected any. "It would be a pleasure. Viscountess Athryn Mithy Arslan! Face me in shuttle bay one of the Sword of Democracy." Ranca virtually purrs her reply, her voice dripping with murderous anticipation.
As the specifics are arranged you pull your ships back away from Chuang Mu and the continual fire of her surface defences, withdrawing into high orbit along with the captured monitor and corvette, sending shuttles with prize crews to secure both vessels then ordering the surrendered orbital forts to evacuate and scuttle. There is a third, around the far side of the planet, but her long ranged torpedo fire is unable to break through the concentrated point defence of six cruisers and a capital ship. Even if some of those cruisers are pointing main batteries at each other.
It takes about an hour before things can be arranged and you take an elevator down from the bridge to shuttle bay one, the huge space largely cleared, pressurised, almost all of your officers gathering as Ranca strides out in her blue and gold armour. Given you opted for a champion, the Viscountess has her cousin, Commander Lord Jose Morgan Arslan, representing her, though she is present and glaring at you quite malevolently through her open face plate.
Ranca skips most of the formalities and gives a perfunctory salute with her blade before then lunging for Lord Jose, who looked shocked.
Ranca
6 + 5 +2 + 17 Prowess = 30
Jose
3 + 6 + 4 + 14 Prowess = 27
Jose Durability 7/10
He responds with literally inhuman speed though as he snaps his own blowing blade on line, catching Ranca's blade near the hilt of his own, a flare of sparks as he attempts a riposte in single time. She counters and the bay rings with the impact of energy blades as they conduct a dozen passes in four heartbeats.
Then Jose is falling back, his expression intent, not acknowledging or feeling any pain despite the melted gouge in the armour at the side of his torso. A slight shudder then as his suit administers combat drugs then Ranca is upon him once more.
Ranca
4+ 5 + 6 + 17 Prowess = 32
Jose
2 + 6 + 3 + 14 Prowess = 25
Jose Durability 0/10, -1 Crippled
This time she opens with a wide, over committed swing, leaving herself open for Jose' answering lunge. A grind of armour boots across the deck as she drops under it then stabs clear through her opponent's thigh, burning right through centimeters of armour then ripping the blade out. The lord staggers even as his profuse bleeding is cut off by his augmentations, crying out briefly in pain, staggering. He looks appealingly to the Viscountess, she gives a curt shake of her head and her faces Ranca back again, this time heavily favouring his leg that has not been half amputated.
Ranca
1 + 4 + 6 + 17 Prowess = 28
Jose
4 + 4 + 4 + 14 Prowess, -1 = 25
Jose Durability -3/10, -3 Crippled
Ranca circles now, blood hissing on her blade before she flicks it expansively to the side, strutting, toying with her wounded prey. Another blurred exchange of blades leads to Lord Jose breaking off minus a finger, he swaps hands, hugging his ruined gauntlet to his injured side, slower.
He does however make no further attempt to yield and Ranca certainly does not encourage him.
Ranca
3 + 3 + 4 + 17 Prowess = 27
Jose
5 + 2 + 4 + 14 Prowess, -3 = 22
Jose Durability -8/10, -8 Crippled
His best efforts do not avail him though and, apparently growing bored as her foe slows, lazily parries his attacks, steps forward, then slices off his arm above the elbow, a downwards flick also hacking into his uninjured leg and almost amputating it after shearing through the armour. Lord Jose screams and falls to the deck with a thud before slowly bleeding, Ranca turns to the Viscountess then smiles so very widely, a casual stretch in her armour, then she deactivates her blade before using its point to scoop up Jose's weapon by the hand guard. She flicks the thing into her hand then offers it to the armoured Captain Athryn. "Your cousin dropped something. Traitor."
Medics are rushing to retrieve the mangled Commander Jose, along with his amputated body parts, they might be salvagable, you intervene before this can go further.
"Viscountess Athryn Mithy Arslan. Your champion was defeated, I am placing you under arrest and relieving you of command over the heavy cruiser Fafnir. I advise you n-"
Athryn is glaring murderously first at Ranca, then at you, then she draws and ignites her sabre. "I do not accept this you Arril bitch. Your puppet will not detain me."
Ranca grins even more widely then without waiting for anyone to make their own intentions known, steps forwards and attempts to push her sabre through Athryn's face.
Ranca
2 + 3 + 6 + 17 Prowess = 28
Athryn
3 + 1 + 3 + 13 Prowess = 20
Athryn Durability 2/10
This does not quite succeed but the Viscountess cries out as she barely parries, Ranca's blade slicing through her helmet and across her cheek, removing half of an ear before they lock blades and shove at each other in their armour. The result is the Arslan heir slamming back into one of her lieutenants as armoured officers hurriedly back away, almost losing her footing, coming back with fast, flurrying slashes as she attempts to land a blow on Ranca.
Ranca
5 +3 + 6 + 17 Prowess = 31
Athryn
2 + 1 + 2 + 13 Prowess = 18
Athryn Durability -11/10 – Dead
So that is when Ranca falls back, waits for a heartbeat until she spots an opening, then steps in to neatly decapitate the heiress to House Arslan, her mangled head falls to the floor. Everyone is silent.
[] Arrest Ranca. She has just committed a crime in killing an Imperial Shareholder, even if it was in a fair duel. She needs to be held in the brig until she can be put on trial upon the return to Ilam.
[] Parole Ranca. Whilst she has technically committed a crime, Ranca is a serving officer and you need her. Have her on parole and 'confined to the ship', not that this will be an issue.
[] Threaten The Arslans. Ranca has just literally cut off the head of their leadership, pointedly accept this then inform the senior officers of the heavy cruisers that they are now under your command whilst she stands looking worryingly pleased with herself at your side.
Regardless of choice I expect you will take cruisers under command, the question is mostly how you deal with Ranca, or do not, letting everyone know that you approve.
Note that the moment Athryn ignited her blade and called Ranca a bitch that was considered a completely legitimate duel, one reflecting poorly on the Viscountess even given that she did not give Ranca time to recover from the previous confrontation. Intervening was very much not an option. Actually killing a ducal heiress though? Ranca is going to be in real trouble.