Somewhat strangely, the best website for planning travel in England and France is often that of the German railways. (That's because it's the only website which amalgamates the timetables of all European train operators):
https://www.bahn.com/en/view/index.shtml
For a sample outbound date of Saturday 27/05, it shows that you can do it with just one change (involving a cross-Paris transfer) as follows:
Depart Ashford International 0655
Arrive Paris Nord 0947
Transfer to Paris Montparnasse (take Metro Line 4 to Montparnasse Bienvenue)
Depart Paris Montparnasse 1228
Arrive St-Jean-de-Luz-Ciboure 1803
(You can actually leave Ashford half an hour later, at 0728, and still arrive in St Jean de Luz at the same time but that involves getting across both Lille and Paris)
[Because only a few Eurostar trains call at Ashford International, you might like to consider departing from Ebbsfleet International instead. It's an easy drive or only about 20 minutes by train]
Returning a fortnight later, you could take the 0940 from St-Jean-de-Luz-Ciboure, arriving Paris Montparnasse 1533. Transfer to Paris Nord for the 1913 departure, arriving Ashford International 2007.
(You can actually arrive 27 minutes earlier if you 1713 Eurostar train to St Pancras International and then go back to Ashford but it's probably not worth bothering with)
The Eurostar website lets you book through fares to many destinations in France but St-Jean-de-Luz-Ciboure doesn't seem to be among them. The SNCF UK website fails to offer any trains for my sample dates
https://uk.voyages-sncf.com/en/
but that might be because it's too early for the fares to be shown. Using earlier dates (in March) suggests a fare of £63 from Paris to St Jean de Luz, and £59.50 back again. The Eurostar website quotes £89.00 from Ashford to Paris, and £36.00 back. So that's a total of £247.50.
http://www.eurostar.com/uk-en
RailEurope also sees the end of May as too far ahead for booking at the moment:
https://www.raileurope.com/index.html
but it could be worth checking later. (NB: Take care when using that website. It uses the American convention for dates, so 05/06/17 is the 6th May, not the 5th of June!)