@mykeal Figured it out: yes, connect the neutral wire in the box to the neutral wire on the HS200; connect the two wires to the gas controller to the two black wires on the HS200.

Unforutnately, this does not work. There apparently is no power to operate the switch; it's dead. Cannot turn on, does not respond to operating the switch. Maybe the line voltage from the gas controller is too low. At any rate, the HS200 does not work in this situation; I'm going to have to find a different switch - anyone have any suggestions?

  @mykeal 

 it sure if your still working on this - but is really easy.  
You need a KASA smart Plug. 
The rest of the parts I ordered from Amazon. 
1 - 3"x2"x2" project box. Amazon $3
1 - 110 VAC Relay.  Amazon $12

1 - AC power cord light weight.  Need the male plug and about 36" of cord.  

Connect the power cord end with no plug to the 2 metal lugs on the relay - that is where you connect the power to.  Should be marked on the relay or box or paper.  
 

Then you need another 2 conductor wire -

like speaker wire - not heavy but fairly light weight. One one end you connect to the two lugs that when the relay is powered up - those two lugs connect.  When not powered up the relay lugs do not touch.  The other end of the wires connect to the two wires on the gas log ingnighter switch. 
 

The short AC power cord gets plugged into the KASA plug.  
 

When you activate the switch it powers the relay which then makes a connection for the lighter for gas logs. 
 

This is all based on the gas lighter being low voltage.  
 

Dor best results solder the Relay connections.  Amazon sells those to (and solder) if you need them.  
 

Might sound complicated - but very easy 

Hope that helps.