Under one interpretation of the rules as written, no feature besides spellcasting from a non-warlock class can be fueled with warlock spell slots, and no warlock feature besides pact magic can be fueled by non-warlock spell slots.
The multiclassing rules for spellcasting tell you to add up your spellcasting slot levels from multiple classes, except for warlock pact magic. It then explicitly lets you use warlock spell slots to cast non-warlock spells, and non-warlocks spell slots to cast warlock spells.
No text permits you to use warlock spell slots on non-warlock class features, and vice versa.
Under this interpretation, you cannot smite using warlock spell slots, you cannot use flexible casting using warlock spell slots, and you do not regain your non-warlock spell slots when you take a short rest.
The alternative, allowing warlock and non-warlock spell slots to be freely used whenever the rules say "spell slots", leads to madness in a few ways.
First, and most famous, is the coffeelock. This uses flexible casting from the sorcerer to sacrifice short rest spell slots for sorcery points, spend them to gain non-warlock spell slots that last until a long rest, and take short rests to regain your warlock spell slots.
A second and even more abusive is the wording of pact magic, which states:
You regain all expended spell slots when you finish a short or long rest.
without the "features always refer only to the spell slots gained by this class" (and the ones combined from other classes), this literally means a wizard 11/warlock 1 regains all expended spell slots from both wizard and warlock when they finish a short rest.
All of this means that the use of "warlock spell slots" in the invocations doesn't matter.
Now, as a DM, you should neither use the "slots are slots" or the "slots can never be used by other class features", but rather deal with it on a case by case basis.
Here bane is a level 1 spell. Permitting you to cast it using any spell slot 1/rest is not going to cause a problem.
Other cases, like polymorph, you might require that it be a 4th level slot or better for non-warlock slots.
Similarly, permitting warlock slots to fuel smite isn't an issue. Coffeelock is.