Another option would be to closed transfer to the purged keg* with ~1-2% extract remaining. It saves the prime addition step but does the same thing. The yeast in suspension consumes the extract it in the keg.
VLB’s Annemüller documents the process in
Managing Fermentation and Maturation of Beer although I think it’s been around longer than that. But he did study the process and document the results as a viable solution. These are the ‘Cliffs Notes’:
https://edelstoffquest.wordpress.com/2016/08/27/fermenting-lagers-in-21-days/ (for an IPA you’d use Ale temps vs the Lager temps cited but the results will be the same)
If you attach a spunding valve you can fairly accurately control keg pressure during this secondary fermentation.
Regardless of how you choose to prime the secondary in the keg, when you do cold condition, I recommend doing it under CO2 pressure. When the temp is lowered, the contents of the keg contracts. As the contents of the keg contracts, you’ll be replacing the void with CO2. If you cool the beer without CO2 pressure applied, due to the design of the keg itself, you’ll draw in air around the seals of the keg to replace that void. This can lead to oxidation. There are many devices used to capture CO2 to use for this purpose but I just use a bottle.
*The keg can be purged with fermentation CO2 to save on bottled CO2.
Below is my 2xBock in secondary now. I am using my fermentation fridge but your closet will serve the same purpose. It started at 1.070. I transferred at 1.014 with an anticipation that this beer will finish ~1.010 or so. Notice the pressure is ~30 psi. (A bit too high IMO) When I transferred to the keg I applied 10 psi from a bottle just to get positive pressure against the seals. The pressure built up to 30 psi naturally and I am getting bubbles in the airlock jar. IOW, the beer is secondary fermenting under pressure. I have considered plumbing another keg into this secondary setup to capture the CO2 but that’s a project still on the drawing board.
**Disclaimer**: Any comment I add is simply the way I brew beer. I am not paid or sponsored by anyone. There are certainly other ways that can be equally effective which other brewers may contribute. This is what I’ve found that works for me using my equipment and processes so I offer this for your consideration. YMMV