Remote timer hack help

Cant see any other posts that are sorta the same as this so here goes, i have made a simple electric chair prop that uses a 12v timer cycle relay board to stop and start bits and pieces of the prop, i was wondering if its possible to link the timer to an older style single push button hard wired remote that i have for a fog machine that doesnt get used much anymore

So in short i want the cycle timer to open its circuit, do its bit with the chair, and activate the fog machine (which is always hot at the mains, awaiting signal from remote)

Cycle timer -
https://www.ebay.com.au/itm/12V-24V...096413?hash=item2cbe2bb4dd:g:fKAAAOSwlPxalMdp

Example of hard wire remote used -
https://gearooz.glopal.com/en-AU/p-....gearooz.com&utm_medium=pr&utm_campaign=en_GB