A dk/sport weight yarn knits a nice-looking piece of fabric using about US4-6 needles. A worsted/aran weight yarn knits a nice-looking piece of fabric using US7-9. Don't try to use a US5 with a worsted weight yarn...it would probably be too crammed and compact looking...like a mat. Conversely, using a US9 with dk weight yarn produces a loosey-goosey holey-looking fabric.
If you are going to try to convert the pattern using a worsted weight yarn...try a US7 needle for your gauge swatch...and see how much wider than 4" your swatch is. For example, the gauge of your pattern probably says something like 22st=4" in st st. By using a dk/sport yarn on US6 needles...your swatch would probably measure 4" wide.
But, if you use a worsted weight using a US7 needle...those 22 swatch stitches might measure 5" wide. That means that for every 22 stitches of your garment, it will be an inch wider.
There are a lot of factors to consider.
However, the short answer is YES...what you are suggesting MIGHT work! You will have a clearer picture by knitting different swatches, measuring the outcomes, doing the math, and comparing your measurements and figures to the schematic of the pattern, or at least to the finished garment measurements given in the pattern.
It is a lot of swatching and figuring...but if you have your heart set on using the heavier yarn, it will be worth it when it turns out beautifully and the right size!