Your entry denial won't affect your fiance visa application. The application might take longer to process because they will have to check up on the details of the bounce, but that won't cause your application to be refused.
You need to draw up a budget showing all the funds you have available to you along with all of your expenses, for your fiance, yourself, and your son. If you have roughly £95 remaining per week after paying for your accommodation, utilities, and transport, (but NOT food and entertainment) then you should be fine.
Also: Do you have sole custody of your son? You will need to show that you have the right to move him out of the US.