Need help with yarn substitution

I have a lace tee pattern that calls for Plymouth Linen Isle. I want to sub it for a Knitpicks cotten yarn but not sure which one is best. The gauge in the pattern is 19sts and 28 rows=4in. I have some Shine Sport I knit a gauge swatch and I get 20 sts in 4in. I tried going up a needle but then I’m off even more. I’m wondering in the Knitpicks Cotlin would be a better sub but I don’t have any to try. Any thoughts?

With the shine sport, try going down a needles size.

What does the cotlin say it’s guage is? if it’s close, you’d have to buy 1 ball to swatch it and see if you can get the same.