In response to the query about this stitch/yarn thickness and 1×1 patterning, 4 ply is too thick for Double Jacquard. Effectively it has become an 8 ply fabric. 1×1 patterning always causes problems with LILI buttons. You get a sort of ragged zigzag. The only solution I have found is to revert to striper backing, or try a double length 1×1 pattern.
4-ply thickness works very well if you use ladder jacquard, with only one stitch in 3 or 4, or even 5 on the ribber. Make the ribber tension as tight as is practical for it to knit off the stitches to avoid ‘tramlines’ on the right side. Experiment a little on a tension swatch to find the ideal.
I think it was Mary Weaver in her book “Easy Fairisle for the Brother Ribber” who introduced us to the idea of ladder jacquard. Subsequent experiments have shown that widening the space between ribber needles in work can produce a lighter fabric, hence Jo’s comment that every 5th ribber needle can be used. Another tip is to set the ribber carriage to slip in one direction which traps the floats every 2nd row. Tramlines on the face of the knitted fabric reduce or even disappear.