Is nucleosynthesis responsible for the expansion of the universe?
Big bang nucleosynthesis did not happen "close to the inflationary period".
inflation happened at about $10^{-36}$s. Nucleosynthisis happened from about 10seconds. 

In relative terms, the ratio of the age of the universe now, to 10 seconds is much much smaller than the ratio of 10 seconds to $10^{-36}$s. And, significantly, nucleosynthesis occurred after inflation, so can't be the cause of inflation.

As for dark energy, nobody has any real model for it. The test for any theory is how well it does empirically. As it stands you have advanced a pet theory, without any mathematical model or analysis.