How is it decided where 了 goes in 这时他得到了公众的支持?
My take on this is that, as you say, the aspectual 了 is placed after the final verb of the sentence, but I'm not aware of the existence of a hard rule for that. I would say it is mainly a matter of style. Maybe there are linguists here who can confirm this.

As for how to use 了 and where to position it in a sentence, this has been asked many times and you can find some pretty good and accepted answers here:
Tense and use of 了


  what is the difference between 这时他得到了公众的支持 and 这时他得到公众的支持了


The first one 这时他得到了公众的支持  is describing a completed action and sets the time in the past --> That time he obtained the support of the crowd.

The second one 这时他得到公众的支持了 describes a change of state (and tells nothing about time) --> This time he's got the support of the crowd. 
Which implies that he usually doesn't have it. 

However, since this last one example describes a change, the sentence with 得到 - which I perceive as a punctual event - plus the absence of a context sounds a bit weird to me.

So I'd prefer the first version any way.