[X] Infrastructure & Surrender. You will limit bombardment of Chuang Mu to infrastructure and take the monitor's surrender. You will seek to provoke the war another way.
[X] Demand A Lot More. Agree to her proposal only if she will make you a baron immediately, with another five shares to follow and a guarantee that you will be given strategic command when you openly move to her families' side in the future. Given people will know where the shares come from, this is going to make it obvious that you are a client of House Arslan, but also a terrifyingly valued one. You would be asking for the value of a planet or nearly two squadrons of capital ships, enough to build up House Arslan's navy by at least a fifth and probably a quarter instead.
[X] Demand More. Agree to her proposal in return for five shares, sufficient to make you a baron, along with assurance of strategic command when you openly move to the side of her family in future. This will make it obvious that you are a client of House Arslan, people take note of that level of share transfer. You would be asking for the value of a pair of battlecruisers with screening fast ships or a flotilla of light cruisers, a significant investment even for a rich ducal house.
[X] Accept Her Proposal. Agree with her proposal, you are being healthily bribed and a single share from somebody as influential and rich as her could be construed as her being genuinely grateful for your assistance and skill in battle, a noble gesture from a mentee to a mentor and attempt to influence you rather than outright patronage from her house.