By creating one resident for the DUF you gain access to it - this is also the place where you can get different shakes!
The Town is fattened up by DIzzy's Delightful Donuts, when you have a branch open and running, the fattening rate can be buffed in the underground facility.
If you use the right home upgrades, a girl will remain with you at your home without ever needing the care home or the facility. They're an alternative gaining route past immobility for girls who don't live with you, or if you don't get the larger bed upgrades in time.
Your best bet might be to fatten up another girl. If you have the Dizzy's shop open & staffed, your employees are good options as they'll passively gain from Dizzy's addiction, plus you can work on the shop floor for workplace fattening shenanigans!