Venting / Negative Pressure - First grow in a tent, in a lung room, both inside a cold garage

Starting my 1st grow in a 4x4 tent inside an insulated lung room. Tent and lung room will be inside a a garage that suffers from big temp swings (which is why I'm building the lung room around the tent - to better control the environment within the cold, or very hot in summer, garage). My plan is to vent the carbon filter / inline fan out of the tent through the lung room and into the garage itself. AC and heater will be stationed in the lung room and humidifier will be in the tent (to keep it a bit separate from the heater).

Is it best to vent the inline fan into the garage itself or outside the garage completely? Temps in garage are cold (for California) in the mornings - 40's to 50's. Further, the lung room I'm building around the 4x4 tent will be small (8'4" long x 6'4" wide x 8' high) so that AC and heater stationed there will move hot / cold air quickly.

Also, will I be able to create enough negative pressure inside the tent with this set-up (from tent thru lung room to garage)?

Equipment purchased:

Here's a rudimentary drawing of what will be my set up:

Gorilla 4x4 tent

AC Infinity T6 + Carbon Filter

Growers Choice ROI E680 LED