That pattern is SOOO cute!
For goodness sakes you should know I am a total beginner at knitting.
So ditch my comments if you like. I recommend that you check with more expert knitters on this question. This is just what I would do for myself!
I might try knitting the same pattern using MY yarn with the recommended needle size for the pattern. But knitting a larger size than I want.
For example - if my kid is a 6, I might try knitting the size 8 or 10.
In this example you would knit out a swatch with your yarn and the recommended needle size using the correct number of stitches.
Because the finer weight yarn will yeild a smaller finished product, you knit out a swatch that is the recomended number of stitches using your yarn.
So if the guage recommended is something like 20 stitches and 18 rows to equal 4 inches. You would knit out a swatch using your yarn and the recomended needles following the guage stiches - 20 across and 18 rows.
Your swatch will be a bit smaller. Now you can compare the difference between how your yarn knits and the pattern size.
To be really safe I would knit a BIG swatch. Double the guage (40 stiches x 36 rows etc)
The idea is to see how much the finer yarn "shrinks" the overall size.
Keep in mind that even changes between sizes are really quite small in most clothes. Usually just a half inch or so or even less. So you might have to follow the instructions for a much larger garment if your swatch is really different in size than the guage recommended.
But if you swatch you should be able to get a rough idea of finished size.
If the size you need is 20 inches in the chest - you will have a gauge repeat of 5. That is 4x5=20.
Using your finer yarn if your swatch measures 3 3/4 inches square - then you will be shy 1/4 each guage repeat. - a deficit of 1/4"x5 = 1 and 1/4" overall.
Then you can compare the sizes listed in the pattern which are usually given in inches at the bust and over all length. So you will be able to see which size to make. If the pattern calls for a casting on 100 stitches for a 20 inch waist (or bust) - but you have a guage deficit of 1 and 1/4 inches - then you can cast on the number of stitches that is required for the size that is closest to that 1 and 1/4 inches you need to make up for.
I hope all of this is making sense!
In any case I wouldnt try using needles more than 1 or 2 sizes bigger than recommended for the yarn I am using. The tension just wont be right and the finished product will be very net like. This might be desireable sometimes. But in a pattern like this one the result will not be much like the pattern you are trying to make.
I hope this helps.