lycopene won't make your skin red. lycopene is a linear tetraterpene without beta-ionone rings, while beta-carotene has cyclic end groups. the rings influence how they insert into lipids and interact with fat cells. beta-carotene’s rings make it more likely to orient in ways that scatter light and change skin hue visibly - lycopene doesn't have this. lycopene is good for skin, but it's effect on hue is negligible.
furthermore, even though lycopene does deposit in fat, it accumulates more in internal organs than in the subcutaneous layer. i don't know exactly why.
even if you're consuming a ton of lycopene and its making it to the fat beneath your skin, human skin is already slightly red due to hemoglobin. red pigments don’t change the perceived color as dramatically as yellow pigments on pale to medium skin.
astaxanthin has the same spectral issues as lycopene(not interacting much with the pigment of human skin.) unlike beta-carotene (non-polar) and lycopene (linear and non-cyclic), astaxanthin is amphipathic. it has both lipophilic and polar ends.
it's even less likely to give you the red tone desired. it doesn't accumulate in the fat beneath the skin, rather things like the muscles and eyes.
the only real way to naturally make human skin red is a sunburn.