A couple of thoughts on this.
First, you are not thermostatically controlled in the tent for that heat vent. So it’s going to fluctuate based on the demand for heat in the living space. Also, if you have a smart thermostat that turns down the heat when you’re out of the house, you are going to have more temperature fluctuation in the tent.
I think using the tent heater with its own thermostat is a more precise way to control things, but if you do decide to pull from the main supply, I’d be sure to install a baffle so that you can turn down (or off) when necessary.