i do wonder how you managed to make nations unhappy that way.
playing the one-base way i also made nations unhappy by the end of second month, but building a second base asap i didn't even come close to getting defeated because of unhappiness, and i tried twice.
first base at lake baykal, because i like how grenades bounce on ice (not even saying that it is a beautiful lake in real life, and therefore an especially good location for the decadent phalanx agents), and second base in nicaragua, third at the northern tip of madagascar (although a better location for radar coverage should be at lake malawi).
so, with more than one base intercepting ufos in the middle of april and cleaning them up quickly as most crashes were close to bases, there seemed to be some 1 to 3 'content' and a few 'happy'.
now i am wondering why you were having trouble, if you say you had few missions, maybe that's just being extremely unlucky or something.
it might mean that, even with the second radar, the aliens were doing their thing outside your radar range most of the time.
and that would be extremely unlucky in my opinion, because in my games, ufos flied around a lot, and eventually ended up in my radar range (looked quite reliable).