You can use any DK weight as long asyou obtain gauge. Switch up needle sizes to find what works for you.
FYI, here’s what I found on that particular yarn that the pattern calls for :
Pearl Cotton has to be one of the most versatile yarns we sell. It can be used for household projects: table cloths, coverlets, dish towels, placemats and napkins, curtains, baby blankets, and scarves - to name a few! It’s also equally at home in knitted garments and woven fabrics for clothing. Pearl Cotton is a 2-ply mercerized cotton yarn with a lustrous sheen and a distinctive twist. It’s easy to work with, strong and durable. It softens with use and has a nice drape when woven in a twill pattern. The best thing about Pearl Cotton is that it comes in 4 weights and a rainbow of over 70 colors, so your options are endless. Buy a lot or buy a little. We offer Pearl Cotton in two put-ups. Halcyon’s exclusive mini cones allow you to buy lots of different colors without buying lots of yarn. The large cones allow you to buy greater quantities at greater savings. Color fastness - Pearl Cotton is generally quite color fast; however, if you are using contrasting colors always test a woven or knitted sample using the same washing method you plan to use on the finished piece. If there is excess dye and bleeding is a problem, skein and wash the yarn before weaving or knitting.
The link to it is here : http://www.halcyonyarn.com/Yarn_pages/yarn082.html
edit at the bottom of the page, it says it’s a sport weight yarn