The device you linked to will work. You are essentially converting to a SPST "thermostat" from your current DPST thermostat. Some people feel a SPST thermostat is dangerous because the heating device is always connected to a hot leg of the power mains. Both legs are disconnected with a DPST device. Yet this SPST practice is common and acceptable in many jurisdictions. It's quite safe, there is no way to contact energized components in a proper installation. And when the heat is on, all components are energized under either system, and yet it's still safe.
The only conceivable danger is if someone assumes it is a DPST system and foolishly attempts to work on the heater without turning off the power at the breakers. If you're OK with this marginal increase in risk, check with your local building authority, ask them if you can use a single pole line thermostat on a 240 volt (assuming US/Canada voltage system) resistive heater. The remaining details aren't pertinent.
If all is OK, the device's installation instructions are fairly clear on how to hook up both 2 and 3 wire low voltage thermostats. Just ensure you order the correct device for your heater's voltage and power or current rating. To be safe, the device rating should be 125% more than your heater's actual power or current draw.
If you or your building authority do not like SPST type installations, you should be able to find a similar relay that is DPST, even if you have to supply the transformer separately. Any possible solution will certainly involve some combination of transformer and relay, it's just a question of packaging and proper ratings.
You can use any heat/cool thermostat on the market to do this. All is required that you run 2 separate 18/4 wires from each system to the thermostat.
R is what carries continuous 24v power to the thermostat. RH is continuous power for heating, and RC is contentious power for cooling. As long as you have constant 24v to these terminals, your system will work. Many modern thermostats have these built in to one terminal on the thermostats (will only show R). All the thermostat does is split the 24v continuous power to RC and RH at there thermostats control board.
From Furnace:
Connect R to R - If your thermostat has RH and RC you can either keep the jumper in, or attach it to just RH for heating. This does not matter as long as you have 24v constant from the unit.
Connect W to W For Heating
Connect G to G for the Fan
From A/c Air handler:
Connect R to R - you can use both R wires at same connection on thermostat if there is only an R terminal. If there is RH and RC like above, you can either leave the jumper in, or connect it to RC.
Connect Y to Y for Cooling
Connect G to G for the Fan
The reason for connecting the G terminal for only the air handler is because you will only really want to have the fan continuously running for the cooling season. Additionally, depending on the type of air handler you have, some control boards require the Y and G terminals to be energized together to run the Cooling and the Fan speed on high. If older units with older style control boards have just Y energized on a call for cooling, the unit will only run the outdoor condenser, which will cause issues with the system. Almost all thermostats will energize G and Y together to avoid this happening if there thermostat is used on a older system.
Best Answer
Simply set your thermostat to "fossil" or "gas/oil" mode
Most thermostats out there support what you describe, as it turns out, as gas furnaces control their blower automatically by default, and only use the G terminal as an override to turn the blower on when the system is otherwise not running. So, simply set your thermostat of choice to "fossil" or "gas/oil", and you'll be set.
(If you're having trouble finding the option, check the installer's manual for your thermostat. Older electronic 'stats had a small switch inside that set this, while newer ones have a software option instead.)