I knew I should’ve given you a stronger dose.
Can someone explain to me how this placement of “hätte” is correct? I thought that with subordinate clauses the finite verb always occupies the final place, i.e. following the infinitive verbs in the verb complex. So shouldn’t it be “…geben sollen hätte”? Is there some different rule for Konjunktiv II that I don’t know about?
There is an excellent explanation by Pitti42 here:
I don’t know why, but this sentence is somehow different, and “geben sollen hätte” does not work for me, and only “hätte geben sollen” sounds ok to my ears. 
Maybe it’s the Konjunktiv II? All my examples in that other discussion are either future, with “werden”, or some other verb, like “wollen” in the simple past. None used the Konjunktiv. Or maybe it’s “sollen” that’s special?
Actually, and this is still weird but a bit better, maybe this kind of works:
Ich wusste, dass ich Ihnen eine stärkere Dosis geben gesollt hätte.
But no one would actually say such a convoluted thing. Especially “gesollt” is a thing you’ll basically never hear (unlike “gewollt”).
If you want to avoid this weird word order, there is an alternative way by dropping the “dass”:
Ich wusste, ich hätte Ihnen eine stärkere Dosis geben sollen.
This sounds very natural and “unproblematic”.
Haha, okay: so this positioning of the finite auxiliary is just a thing you are allowed (but not required) to do only with auxiliary verbs and only in the special case where the verb-complex has more than two verbs.
Just when you think you’ve got it sussed out, there is another curveball. 
Thanks!
Yes, that last formulation sounds less awkward than any of the rest. I know that in practice there are ways to avoid this kind of thing, I just wanted to understand what is going on grammatically. Good to know that this one is a little obscure even to a native speaker. 
Thanks!
Just following up on this in case anyone is curious, I have learned that the technical term for this construction in German grammar is Oberfeldbildung and it is mandatory if, and only if, the auxiliary verb in a subordinate clause would be applied to a modal verb in the so-called Ersatzinfinitiv construction (commonly called “double infinitive” in English).
So dass er das Fahrrad hat fahren wollen is correct while dass er das Fahrrad fahren wollen hat is not, and dass ich sie spielen gesehen habe is correct while dass ich sie habe spielen gesehen is not.
And because some other verbs like lassen can act as quasi-modal verbs, there appears to be some wiggle room with them: apparently dass sie ihn hat gewähren lassen is acceptable and so is dass sie ihn gewähren lassen hat.
Alles klar? 
(Of course the natives should correct me if I have screwed this up!)
If anyone really wants to dive deep after that, I found this article that contains more than you ever wanted to know about this quirk of German grammar. (Warning, it is not light reading if you are DaF!)
Man, this PDF really is “starker Tobak” and hard to understand 
To the point that the english version of the abstract was easier for me to understand, and the german is basically “technobabble”.
I mean what is meant by “zentripetal organisiert”, “Oberflächenprofil”, “Peripherie”?
“Oberfeldbildung” itself is pretty obscure (“formation of an upper field”? ) until you come to the end of page 247 where there’s in parentheses (“Verb-Cluster-Bildung”) which makes it clear.
Props to you if you actually understood all that.
I would say I only understood 40% of it, but that was enough! 