Leave the old paint on. But go over all of it with very fine sandpaper, this will roughen the old paint sufficiently for adhesion of new paint. Take it to a pro who has lots of experience painting composites (usually this means a bike painter, but some high-end body shops have experience painting carbon fiber car parts). They will clean it very thoroughly with composite-compatible solvent. Paint as desired.

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Advanced Aero TopTube Storage for Road, Gravel, & Tri...ZeroSlip & Direct-mount, made in the USA.
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Use someone who is good at painting carbon. It's much more complex than painting alloy.

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http://www.falcobike.com
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Thanks just what looking for