[X] 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.
[X] Arrest Her. She has no right to challenge a senior officer to a duel during active combat, it is arguably illegal. Hard dock the Sword of Democracy with her cruiser and arrest her. You have a faster ship and a vastly larger marine complement, they would be suicidal to resist. This could get messy if her other ships intervene but things are still overwhelmingly in your favour.
[x] Duel Her. Probably not very wise, her record states she is pretty dangerous with a blade. If you win then you can probably take her into custody without any issues and have retroactively justified your accusations, losing however would be very bad even if she does not kill you.