It looks highly complicated and unpractical.
First of all, you are stitching together multiple systems which are supposed to work one after the other, and an old motto among reliability engineers states that
What is not present cannot get broken
Let's then give a look at some basic physics equations.
The distance covered starting from still can be calculated via
$ d = 1/2 a t^2$
while the velocity can be calculated as
$v = at$
Based on your constraints of $d = 18 \ km$ and $v = 150 \ km/s$, I get an acceleration of about $600 \ km/s^2$ and a time of about a quarter of a second.
That means 60000 times the acceleration we experience on Earth. If you push 1 kg with that acceleration, the projectile will experience a force equivalent to the weight of 600 tons!
I have an hard time figuring out any precision mechanism that can withstand such forces without breaking apart. And after that smack, you want also to use the other two stages...
My advice would be: go simple. 1 rail gun on hormones is already enough for the suspension of disbelief. Don't challenge it further.