Old player recently returned. I have been doing class stories I have never done and noticed the influence for those companions goes up pretty nicely.
For my old BH though having Mako at just 10 influence at level 55 is painful. For a test, I just ran Black Talon solo and got 4560 influence in about 20 mins. The bad side is a picked two light side convo choices (+200) which did not match my BH because I wanted to see the max influence.
Not sure if there is weekly lockout on that, but will run again without choosing light on those two and see how much influence I get.