The Melancholy Funk:

• The Sad Trombones
• Blue Note Blues
• Gloom Funk
• Velvet Tears
• Midnight Groove
• Soulful Sighs
• Rainy Day Rhythm
• The Weeping Willows
• Downcast Groove
• Melancholy Moaners
• The Somber Souls
• Apathetic Funk
• The Disillusioned
• Static Sighs
• The Uninspired
• Grey Funk
• The Listless
• Monochrome Groove

The Apathetic Iconoclast:

• The Uncaring
• The Rebellious Slumber
• The Indifferent
• Iconoclastic Echoes
• Apathetic Rebellion
• The Broken Idols
• The Unmoved
• Static Resistance
• The Disengaged
• The Apathetic Revolution
• The Silent Protest
• The Unfazed
• The Lethargic Rebels
• The Numb
• The Unconcerned
• The apathetic icon
• The indifferent iconoclast
• The apathetic rebel

Combined Names (Melancholy Funk & Apathetic Iconoclast):

• The Melancholy Iconoclasts
• Apathetic Funk
• The Sad Rebels
• The Gloomy Iconoclasts
• The Uncaring Groove
• Melancholy Resistance
• The Indifferent Funk
• The Apathetic Sighs
• The Rebellious Blues
• The Unmoved Groove
• The Disillusioned Iconoclasts
• The Static Rebellion
• The Listless Iconoclasts
• The Uninspired Rebels
• The Grey Rebellion
• The Numb Funk
• The Unconcerned Groove
• The Lethargic Iconoclasts

This list provides a variety of options, playing with different aspects of the input keywords. Remember to check for trademark conflicts before settling on a name.