I think
$a=3$, $b=12$, $c=12$ works well.
Check:
Sum of all the edges' length: $4\times(a+b+c)=4\times(3+12+12)=4\times27=108$
Surface: $2\times(a\times b+a\times c+b\times c)=2\times(3\times12+3\times12+12\times12)=$
$=2\times(36+36+144)=2\times216=432$
Volume: $a\times b\times c=3\times12\times12=432$
$3:12:12=108:432:432$
How to find this:
The four equations can be written as:
$4\times(a+b+c)=k\times a$
$2\times(a\times b+a\times c+b\times c)=k\times b$
$a\times b\times c=k\times c$
From the last one, we immediately get $k=ab$.
Using this on the second one, we get $c=\frac{a^2 b}4-a-b$.
All this turns the first equation into $b^2(-\frac{a^2}4+\frac{a}2+1)+b(-\frac{a^3}4+a)+a^2=0$
The square root of the discriminant has to be a rational number, so $a^4+8a^2-32a-48$ has to be a perfect square.