Except where noted in the multiclassing rules, when you gain class features depends on your class level. In this case, invocations are class features that don't have special rules in the multiclassing rules that say otherwise, so you get them according to your class level.
Note that in the case of invocations, it's even stronger than "it's not mentioned in the multiclassing exceptions": the rules for invocations specifically call out your warlock level (PHB, p. 107, "Eldritch Invocations", emphasis mine):
When you gain certain warlock levels...
when you gain a level in this class...
If that isn't clear enough, the specifics of how you gain and choose invocations directly references your warlock level (all emphasis mine):
When you gain certain warlock levels, you gain additional invocations of your choice...
This next quote is especially telling, since it explicitly refers to the rules' notion that the current warlock level is what determines which invocations you can learn, and would be nonsensical if character level is what mattered:
Additionally, when you gain a level in this class, you can choose one of the invocations you know and replace it with another invocation that you could learn at that level.
Everything about invocations references warlock level, not character level.