Where is a good spot to find connectors and crimping tools for this smaller cable? The picture is something like the connection on the existing cable.
BNC connector looks like a twist on so no tools needed, many electronic outlet stores will make up cable per order, RG-8X is the same as RG- mini8, be careful to get the right shielding and center conductors when ordering, it comes in solid, stranded, stranded silver tinned center, and or both silver tinned center and silver tinned shield, I just ran a SCADA antenna feed with 8X and use the MIL spec silver tinned center and shield for a sewer transfer station I wired, a local electronics store made the cable up for me just for the price of the cable and ends, so most don't even charge to make it up. they can also sell you the tools to make your own up.
Just make sure your allowed to do this work, as these detectors are required to be certified, we could only wire up to them and provide conduits, as far as the units go they have their own installers who do the rest, so make sure you ask about this, as they are licensed through the government.
RG-8X is just about the same size as RG-6 so that sounds about right for your need.
you can get the cable on line but most places want you to buy in bulk, not good for just one application.