ITA it will be a lot less hassle just to fly straight from Southampton to Paris. 15 mins taxi to the airport, then a 1 hour flight. Even arriving at the airport 1 hour early, makes the total journey only 2hr15 mins, and no baggage hauling!
for Southampton flights.
By train the journey would be 1hr20 mins from Southampton to Waterloo. Then 30 mins on the tube to Kings Cross/St Pancras including at least one change of train. But you would have to make sure you aimed to get there at least an hour in advance of your train (in case of delays) so when you add on the train journey to paris (say 2hr30) you would need to allow 5hr30 to make this journey at least I reckon, to make sure no connections were missed. Given the sheer grimness of taking any luggage on the tube, you could choose to take a taxi from Waterloo to St Pancras instead.
If you are flying home from London you could always come back on eurostar. The tunnel is not much of an experience. It's just dark!