Train from Bayeux to San Sebastián

Yes, 4 changes, vs 2 if you change stations in Paris, which I suppose makes it 3 changes. They both take exactly the same amount of time.

No, not direct and it would take a fair effort to bypass Paris. The best connections go to Paris and require changing from Paris St. Lazare station to Paris Montparnasse station by metro or taxi. They reach the border town of Hendaye, France in 8-9 hours, see schedules. From there, the local Euskotren provides frequent service to San Sebastian, 35 minutes more.

Also look at www.skyscanner.com to compare flights between Caen and San Sebastian (or Biarritz or Bilbao).

Bayeux to Paris to Hendiaye (change for the Euskotren https://www.euskotren.eus/) to San Sebastian

No hope of getting a direct train for a trip between two such relatively small places at opposite ends of the country. You will have to change trains in Paris.

To find train times (but not to buy tickets), go to the DB website: https://www.bahn.com/en/view/index.shtml
Enter; from Bayeux, to San Sebastian, and desired date and time.
I tried a random date, and got a 10½ hour trip:

Bayeux dep 08:35, Paris St Lazare arr 10:49
Cross Paris by métro
Paris Montparnasse dep 12:52, Hendaye arr 17:31
Hendaye dep 18:35, San Sebastian arr 19:08

3 trains plus métro across Paris.

Hi,
While there is a way to avoid Paris on that trip, as said above it involves 4 changes, so I would worry about its reliability. For documentary purposes, you can do:
Bayeux-Caen
Caen-Saint Pierre des Corps (near Tours)
Then TGV Saint Pierre des Corps-Bordeaux
Bordeaux-Hendaye
Euskotren Hendaye-San Sebastián

Compared to the alternative of changing trains in Paris it does not save time. It might come slightly cheaper but only slightly. And the change of trains in Paris is not hard, there is a direct metro (line 13) between Saint Lazare and Montparnasse stations, with shortish walks (with stairs). 90 minutes between trains allows for a safe connection.