I'm no cyclist and others may have better advice, but as no-one has replied and I've done both routes, I'll proffer some information.
April 13 is very early in the season and the weather very chancy. I'd wait to decide which route until at SJPDP and then take advice from the helpers in the pilgrims office. If the forecasted weather is at all unfavourable, then take the Valcarlos route and stick to the road. Bad weather in the Pyrenees can be life threatening.
There is a pathway that shadows the Valcarlos road, but after Valcarlos it would be very difficult I think, in early April, as it is likely to be wet - so muddy, narrow and slippery. It runs up and down galleys and through beech forests.
A lot of the Napoleon is on a minor road up, then one section a track across fairly open country, and then narrow (and can be muddy) along the border fence, then the last section where there is a choice of going down a very steep slippery slope through the beech forest, or taking the slightly longer minor road.
If you are lucky and the weather is bright and sunny, you may be fine on the Napoleon - but the walkers will not be impressed when you try to pass on the narrow sections.
Take care and enjoy!