I plan to travel out on the 26th April, returning 10th May.
I'm heading for Tuscany, then going to Milan on the 5th for work.
Transport in my reliable and fully insured VW Transporter - T4, comfortable seats and air con. Room for you, your bike and luggage.
Probably a two day drive down/one day back. Split the driving (you have to be >25 or provide your own insurance) and Eurotunnel/fuel/tolls cost. Maybe £125 plus accommodation (if necessary) and food, less if you only want part way.
I could shift the dates by a day or two (except the Tuscany/Milan leg, if you wanted that) and the route is a moveable feast - if you wanted to go to France/Switzerland/Germany etc., that would be fine.