thanks to the happiness figures that guy on reddit posted i figured you should be able to get 100% with 2 settlers, which would be good cause i could never be bothered with the 7 settler way thats been documented
(i did this previously but got it wrong and built med lab instead of bar as i though that gave more happiness doh and got stuck on 95% which made me think i was wrong and its not possible , was meaning to post this but forgot then something i read reminded me)
so with 2 settlers all you need is enough food water and defense to cover and a level 3 bar
so say 3 water (1 pump)
1 settler working on food crop, 6 food
2 heavy machine gun turrets (16 defence which is higher than the 9 food and water production)
2 nice beds (not sure if they have to be the clean ones? but i only build those) under shelter
1 settler working in level 3 bar
thats it 100% happiness
so all you need are the resources for 1 water pump, 6 foods worth of crops, 2 beds (and shelter if required), 2 heavy machine gun turrets and a level 3 bar
the bar is cheap doesnt require many caps to build a mere 1500
you will need level 2 local leader and level 2 cap collector as well
with this i got the achievement at hangmans alley, i never had to wait there or sleep there or hang with the settlers or anything it just increased by itself over time no matter where or what i was doing, and no decorations built (as they dont do anything)
although i had 6 food 12 water (for extra) and maybe 64 defense (for game 4 turrets covering each entrance ) and 3 beds, but you will only need the minimum food, water, beds and defense required
edit: oh forgot to say this was pre patch, i have no idea what "changed happiness so it calculates correctly now" means under patch notes and whether that would not effect this or make easier/harder etc