Narrowband LED performance compared to Laser for Spectroscopy
The three requirements you listed are what every designer wants: low price, high performance, low power. My electronics professor used to say that The designer's blanket is always short: if you pull it up to cover your face your feet will be cold, and vice versa.

There are two ways of deciding on this matter:


Budget constraint
Specification constraint


You have to decide what's more important, then you pick your components.

I'm not an export in Raman spectroscopy, but I assume that there is a relationship between the laser bandwidth and the amount of information you can recover from this technique.

If you decide that cost is paramount you'll be forced to use a laser diode since those are cheaper. If you have a price range, say $100, you can start to shop around and try to find the best laser diode that $100 can buy.
When you find what you want, you design your system around the specific diode characteristics.

If you decide that performance is all that matters, you will do the opposite thing. Let's say that you calculated that in order to meet your specifications you need a light source with a bandwidth of 1nm. At this point you can shop around and find the cheaper laser that gives you that bandwidth. Again, at this point you can design your system around it.

This argument obviously applies to every component in your system. It wouldn't make sense to have a very expensive narrow-band laser and then use a cheap monochromator that would kill your information.

If, as you said, price is the number one priority, go with the laser diode.