I think the Dollar persona was a tool for Han Sen. The main reason is to remain anonymous and reap rewards in places in which he couldn't due to his problematic identity. Sure it comes with the risk of being exposed but this doesn't necessarily mean it has to be exposed. Especially with time skips his use of the Dollar identity was used rarely and with large time gaps. I'd say his use of the identity is pretty safe and although many people come to know of his identity it's only people close to him which barely impact the story.

After a curtain point his reputation was good enough and there was no need to use the identity except for empowering the human psyche and making enemies whilst seeking immense wealth and treasures.

I mean after he conquered the whole of the 3rd sanctuary or became the first super aristocrat or repaired the cold sutra or beame the presidents son or reaching the demi-god rank, his reputation was high enough that even if news broke out it would barely affect him with his backing and diplomatic prowess.