UK Visitor Visa Refused Due to Income Mismatch and No Right to Appeal - Need Honest Advice

I recently got refused for a UK Standard Visitor Visa from the USA and wanted honest advice before deciding whether to apply again.
I am a Pakistani passport holder but a US permanent resident (green card holder). My wife is a US citizen and we both live and work in New York. We applied to visit my sister in the UK and also do tourism for around 15 days.
This was my third application. My first refusal in 2024 was expected because my application was weak at the time. My second refusal in 2025 happened because I did not upload documents properly. This latest refusal confused me because I thought my case was finally strong.
I submitted:

Green card

Bank statements

Paystubs

Tax returns

Driving license

Marriage documents

Invitation letter from my sister

Cover letter

Salary increase letter

Wife US passport copy

The refusal basically said:
They could not fully verify my relationship/contact with my sister. (Even though I gave them her passport number)

My declared monthly income did not match the amount of money entering my bank account.

The problem is that I have multiple jobs and income sources and earn a very good amount of money overall. While filling out the application, I added my main job, but I genuinely could not find a proper section or option to clearly explain my second and third jobs/income sources. Because of that, I only listed one job on the application, while my bank account reflected income from multiple sources and transfers.
I think UKVI assumed the extra deposits were unexplained income, even though the money was legitimate and connected to my other work.
I also received a notice saying there is no right to appeal or administrative review for this refusal, which was disappointing because I feel the issue was more about misunderstanding and lack of explanation rather than anything fraudulent.
I have no immigration violations, no overstays, strong US ties, and my wife is a US citizen. We genuinely only wanted to visit family and travel.
Do you guys think this is fixable if I clearly explain all income sources next time and provide birth certificates/family proof? Or does having multiple refusals make future approvals very difficult?
Would appreciate honest advice from anyone with UK visa experience.