The description for that item says it works for your bard spells and thus it only works for bard spells. This is one of the core principles of D&D 5E - everything does what their description says they do.
If you had the same spell as a bard then you could add the bonus from the drum but you can't use it for spells from other sources. If you could then you could arguably stack an item that increases warlock spell DC with one that increases bard spell DC.
It also doesn't matter what spell slot you use - there is no real difference according to the rules:
Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, and half your levels (rounded down) in the paladin and ranger classes. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.
Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.