Hi Ornella! Great idea to keep the scarf but change the color to work with your wardrobe.
While I have never dyed a silk scarf (nor anything else made of silk), in the past I have dyed loads of cotton items - tops, sweaters, jeans, baby onesies, pillowcases, etc.
So this is based on my experience with dyeing cotton items:
You're right that you'll need to plan for a darker color if you're dyeing over another color. A rich red would probably cover the lavender nicely - but the underlying lavender may affect the exact hue of red you get - it may be a purplish red. The lavender probably has blue and gray in it as well as a bit of red, so that is likely to affect your final color. The paler your lavender is, the less it will affect your final red color.
Most dyes work best on natural fibers. So silk and cotton are good candidates. But if your scarf is a silk blended with a non-natural fiber (like polyester), you'll get a different effect as the silk takes up the dye and the other fiber may take up little (if any) of the dye. That can still be a beautiful effect, though!
If you have a choice between powdered dye and liquid dye, go for liquid. Powdered dye can be futzy. If it isn't completely dissolved before you add your fabric to the mix, your finished scarf can wind up with undesirable blotches of darker dye.
Be sure to follow the dye package instructions exactly. Most say to have the garment thoroughly saturated with water before adding it to the dye pot. If that's part of the instructions on your dye, be sure to do so - it will help your piece turn out evenly colored.
Also if the instructions say to stir or agitate the garment for the full dyeing process, be sure to do so. It will help your finished dye job turn out more evenly.
Part of the fun of dyeing things is seeing how they turn out! It's best to be open to the color result and not expect an exact hue.
Good luck, Ornella, and I can't wait to see your scarf after its makeover!