You can easily expand the armhole at this point in the pattern where the original armholes are described:
“k3, bind off 6 sts, k to 10 sts from end, bind off 6 sts, k3
- p3, cast on 6 sts, p to bind off sts, cast on 6 sts, p3”
You can bind off more than 6 sts if you think necessary and instead of casting on the same number of sts in the very next row, knit as many rows as you think necessary and then cast on over the openings. In order to do that, you’re going to have to purl to the cast off sts, start on the middle sts with a new end of yarn and then purl the last sts with a third end of yarn. This is a common way of doing a dog sweater.
You might just try a swatch first with some rows of knitting, a 6 st cast off and the cast on in the next row as directed in the pattern just to see how large that opening is and to get an idea of how big the armhole should be for your dog.