I guess the answer to your question depends on what size garment you are trying to make. If you need the smaller size, then, of course, you'll need to use a larger needle to get the right gauge. If you want to make a larger size, you could use the size 5 needles and follow the smaller sized pattern (since your gauge is more stitches per inch than what is recommended).
Personally, though, I'd try to get the right gauge by using a larger needle. That way you'll have a pretty good idea of the final size of the garment.
Also note that in most patterns, it's more important to get the number of stitches per inch than the number of rows per inch. Most patterns allow you to add rows until you hit the right length.
Can you give us a name and/or link to your pattern, please?
Good idea to try a larger needle size. More stitches per inch will make a tighter fabric and that's not what the pattern is calling for. It could make an appreciable difference. If I thought I were more likely to make gauge by going up 2 sizes, I'd try a needle 2 sizes larger first. It all comes down to trying whatever sizes until you get the one that works. If you're working cables, looser is better.
I looked on real very and it is not listed. I tried it with a 7 needle and 100 stitches (vs a 5 needle and 94 stitches)and it is still too small. So strange that I could be so far off. I'm even using the yarn recommended in the pattern.
The gauge is pretty small for size 5 needles - are they 5mm/8US -maybe? That would give you about 5 sts per inch. Always go up in needle size until you match the stitch gauge; the row gauge isn't quite as important as lengths are usually given in inches. Rather than CO 100 sts, try about 25 sts and knit for 3 or 4" and then measure your sts per inch, a larger piece will give you a more accurate measure.