Hi Chris,
I did a diagram of an approach for you, before the above posts came in.
It is based on my interpretation of what you initially said and also my assumptions about the timer module function. Please don't treat it as a final design because there are a few things to confirm before we can reach that stage. Instead treat it as notional and for discussion. The diagram shows the Timer Module in the REMOTE LOCATION, of course there is no reason why it could not be in the LOCAL LOCATION.
From the picture on the link you posted, you can see that the relay can handle 10A, but it could be that the igniter takes more current than 10A, especially when starting up so that is one area to watch.
One final point: would you be able/prepared to build a simple circuit from scratch? If so we, could do pretty much anything within reason.
The arrangement shown in the attachment works like this:
(1) With the switch open '12V PULSE' will be at 0V.
(2) When the switch is pressed '12V PULSE' will immediately jump to +12V.
(3) After a time, that you can set with a variable potentiometer on the Timer Module, '12V PULSE' will return to 0V.
(4) Whenever the switch is open '12V PULSE' will always go to 0V, regardless of the state of the Timer Module.