
Well... the problem with that is that che chance of it getting to those people ears is too big. Remember that at first all servants and knights were against her... and even her building the lounge and going to the training grounds got reported to her father. If she had done that and they found out it would've been all for naugh.
Honestly the biggest flaw I see in this is her father that cares so much about his family's image bringing in an illegitimate child into the house... like... did he thought no one will talk about it? Wouldn't it be considered especially shameful after the previous cp fiasco? That feels like such a big plotholes for me...

I think I was predisposed to the FL being a lone-wolf CEO by the premise of the story: FL kicked out of family; starting from the shadows raises herself to be the wealthiest person in the empire. I was not expecting a shadow investor, but, now that I've thought about it FL most likely would have needed some help to get started even if it was a little. I haven't gotten to her trying to get on the knight's good side, yet. But no worries -- spoilers won't stop me. ^_^
However, you do bring up an excellent point about her father. For a nobleman or head of household to bring in a bastard/illegitimate child to raise when he already has two healthy legitimate children is unheard of in these types of stories. Especially when he's is overly concerned about how the family is perceived by other nobles and royalty. Since it hasn't been explained, unless I overlooked it, perhaps, the father brought the youngest child into the family fold due to her exhibiting powers gifted from his bloodline?? Was he looking for healing to manifest??

Omg I'm so sorry... giving spoilers was not my intention... my apologies...
I used to want them to not depend on others too... then I remembered just how little autonomy women used to have in the past... and it bagins make sense how no matter how strong or intelligent mc is she usually need a man to help her.. either as a business partner or even being a figurehead in her place.
It's really weird isn't it? Even more so when his wife is alive and well... I also think it's probably because of the gift of their bloodline... but I still think it would make more sense if he kept her away and brought her in after her bloodline awakened... honestly I'm torn in regards to illegitimate children. On one side it's not their fault to be born right... but on the other it's very disrespectful to the wife and legitimate children to bring them in and try to force them into the family. His wife is a saint... unfortunately the girl is a bish and the son is too dumb... guess he takes after his father.

Selling things would raise suspicion and technically they have to report all expenses in the books (I mean the household, not the business).
Having an investment partner who can keep identity secret and provide funds that her family can’t trace is a smart businesswoman move
No different to the real world. She is making it on her own even by getting a business partner. It’s not a sign of hopeless damsel
I just started reading and I'm a few (or more!) chapters in and I'm slightly disappointed that the FL didn't start her cosmetic company on her own. I know she needed some capital and the best way to get it is through an investment, but she's a duke's daughter -- can't she ask her mother or sell some jewels/accessories/shoes/dresses to get the cash flow needed toward starting her own company??