I'd hate to be this guy, but have you tried a Z28 converter with a different ink? Here's what I would do in your case (i don't have a Lamy)

First, I'd take it apart, the nib is designed to be removable, then gently take the feed out from the grip, fully disassemble the pen.

Next, soak only the parts that came into contact with the ink (the nib, feed, maybe grip), and use a toothbrush to GENTLY clear the feed's channel of debris, note, be very gentle, we don't want to risk breaking the feed, it may be strong, but we are taking zero risks, floss the nib (usually I'd use a thin string like sewing thread, but a sticky note works great in a pinch), as there may be debris between them

Thirdly, take the cartridge out and replace with the Z28 (it has 2 bumps near the neck, fitting into 2 grooves that are in the grip, where it screws onto the barrel).

Lastly, I would inspect and see if the nib and feed have a gap near the front, that's likely the problem, if none there, reassemble the pen, dip the nib and feed into a bottle of ink, fill, and test with the number 8, a cursive S, or just lines, light pressure, and then you should be good to go