Question regarding Pact magic with multiclassing and a class specific bonus

I play a 6 bard/ 4 warlock multiclass. My character recently obtained a rhythm makers drum +3, giving a +3 bonus to my bard spells’ DC and rolls.

I am curious if I were to cast a spell I had taken on my warlock list (hold person) with a bard spell slot, would the bonus increase the DC? DnDB added the bonus to all my characters spells (I know that isn’t right) but thought it may be applicable if a spell is on both lists.

Thank you in advance.