You can call latexmk via AUCTeX which with some minor configuration can compile LaTeX with Asymptote figures.
According to the Asymptote manual you can use latexmk to compile LaTeX files containing Asymptote figures with a single command (ยง7):
An even better method for processing a LaTeX file with embedded
Asymptote code is to use the latexmk utility from
http://www.ctan.org/tex-archive/support/latexmk/ after putting the
contents of
http://asymptote.svn.sourceforge.net/viewvc/asymptote/trunk/asymptote/doc/latexmkrc
in a file latexmkrc in the same directory. The command
latexmk -pdf latexusage
will then call Asymptote automatically, recompiling only the figures
that have changed. Since each figure is compiled in a separate system
process, this method also tends to use less memory.
What this means is that if you add the following to your latexmkrc
$pdf_mode = 1;
sub asy {return system("asy '$_[0]'");}
add_cus_dep("asy","eps",0,"asy");
add_cus_dep("asy","pdf",0,"asy");
add_cus_dep("asy","tex",0,"asy");
and run
latexmk your_document
it will do all the necessary steps to produce a PDF of your LaTeX document with Asymptote figures. Such a command can be invoked via AUCTeX to do what you want, i.e. latexmk can be called from AUCTeX.