After some extensive testing..8 hours of it!
You only need 1 settlerI have a much easier solution which will take a few hours maybe 3. While clearing red rocket to redesign and rebuild..I noticed happiness increased as I sent settlers away. (Indicated by the arrow alongside the happiness bar)
I suggest a manual save before doing this as you burn a lot of building materials.
NOTE: DO NOT LEAVE THE SETTLEMENT ONCE YOU BEGIN! HAPPINESS WILL DROP.
The criteria to meet:
1. Remove Radio beacon if you have built one
2. Reduce the settlement to 1 settler (Send all others to settlements by pressing
while in workshop mode and selecting any settlement.
3. Build a Restaurant - Third store under food and drink.
Requires cap collector lvl 2 and Local Leader lvl 2 perks. and 1500 caps.
4. Place 2 sleeping bags inside the red Rocket building (little room in the back is ideal)
5. Build 1 Chair
6. Large settlement: To Achieve a large settlement - This is not the number of people but items in the area. You need to keep placing items until the size bar goes Yellow. Radios worked well for me just spam them everywhere. Just keep building anything you can.
7. Food and water must be present within the workshop (Settlers can use this in place of a pump/food. I had around 50 purified water and a mixture of crops totalling 50.
Once the above is in place:
1. Wait until 10 am on the chair
2. Get off the chair and wait about 20 minutes (Periodically enter workshop mode to check the happiness rating. If the happiness rating does not increase by 14:00 (2pm) move to step 3.
3. Once the happiness rating has increased. (It will rise by one or two points). Wait on the chair until 10 pm (doesn't have to be exact).
4. at 10 pm sleep until 10 am
5. Repeat the above until you get the trophy!
I got this 5th May 16 with latest patches. Took around 3 hours to get!
Happy Hunting