I have a long (28") plastic spoon. Sometimes my conical gets clogged with highly flocculating yeasts. So I sanitize the spoon and give it a good stir. I've also used this method to rouse Hefeweizen yeast to get the cloudy suspension.
I also have an old refrigerator box with a 100 watt light bulb connected to a inkbird temperature controller. The light bulb is wrapped in foil, so it's essentially a heater. I tape the inkbird temperature sensor (using insulated tape) to the side of my carboy or conical. It can easily maintain 90°F in my nominal 65°F basement. I normally use it for a diacetyl rest (80-85°F), but it's a great way to kick-start a stuck fermentation.
Making a rig to inject CO2 from the bottom seems unnecessarily complicated. You'll need to sanitize whatever adapter you fabricate, the mating threads when you assemble it, and the tubing. You might get some backflow into the gas line when you open the racking valve. psi is irrelevant. You just need enough pressure to overcome the hydrostatic pressure at the bottom of the conical, then it becomes a flow problem. Simply adjust the regulator until you see bubbles.