My vassal bishops are paying taxes to the Kaiser of the HRE and not to me, even though they like me significantly more. I can't understand why this is.
When the pope is a vassal of an emperor, bishop vassals pay taxes directly to the emperor, bypassing the pope; just as bishops pay the liege of an antipope rather than the antipope.
However, in what appears to be an oversight, the UI shows the bishops opinion of who he's paying, not the opinion of who governs who he's paying (iirc this has been the behavior since launch 2 years ago).
You can get the actual opinion of the vassal pope or anti-pope in a very roundabout manner. As you can see here, the displayed value is of Guilhem's opinion of the HRE Heinrich:
If we bring up the diplomacy menu of Antipope Vitalian and click on this button to bring up other's opinions:
Then we can see that Guilhem's opinion of Vitalian is higher, and thus he is correctly paying his dues to Heinrich: