A couple of friends of mine regularly take expensive road bikes on U2 to Faro, for what it's worth, and are pretty impressed by the bike handling (LGW-FAO).

If you want to take the bike as-is, you can't book a through cycle reservation.

For a through Eurostar ticket, you'd have to contact Eurostar separately and check there is space. Bikes get sent as registered luggage (£25 on the day) although you have to specify if you want it to be delivered on a specific train. Space is limited, particularly in summer. You'll then need to contact SNCF offline to pay the €10 supplement to take the bike on the TGV.

If you book a separate SNCF connection online (e.g. via Lille - which would be a lot less hassle than Paris) then you can reserve a bike space while you're booking on voyages-sncf.com

If you were doing the bike shuttle, you'd need to arrange for a taxi that could take your bike (plenty will) on the Calais side. You couldn't walk that.

Otherwise, invest in a bike bag - plenty of them on eBay. If you can dismantle your bike and place inside a bag, and it has dimensions of under 85cm, you can take it as luggage on both Eurostar and TGV, free of charge. It's fun and games at security, though...

Another option, if you're going the ferry or Eurotunnel bike shuttle route - you could invest in a trailer - again, eBay is your friend - to allow you to take a decent sized case with you. It'll also help with the annoying station-port transfer in Dover.