Travelling from Bilbao to Bordeaux on September 22nd. What would be the fastest way to get to Bordeaux on a Saturday?

Airplane.

I take it you're not driving. If so, then bus to San Sebastián, connecting to Biarritz train station. Train from there to Bordeaux.

"fastest way" was asked for and that bus and train would take how long? Several hours I'd wager.

7 hours, 30 minutes to take the bus to Donostia, then to Hendaye, then to Bordeaux. Obviously, plane would be faster.

We made the same trip you suggest, Bilbao to San Sebastian by bus (for a few days), then train to Bordeaux. It required that we change trains at the border by walking across to the French train station. The two stations were very close together, perhaps a hundred yards. I don't know if it is a difference in train gauges, or some other problem, but we could not continue on our Spanish train through to France. From the border station (not sure of the name), we traveled on to Bordeaux.

The border station is at Hendaye. The train gauges are different; hence the need to walk from one country to the other to switch trains.

But it's a long day on public transportation. Take a plane.

Difficult to fly from Bilbao to Bordeaux as there are no direct flights. You would have to take a connecting flight through Paris or Barcelona.
Airport destinations - Bilbao Airport - Aena.es

The ALSA bus from Bilbao to Bordeaux takes about 4-1/2 hours, with a change of buses in San Sebastián-Donostia.

4.5 hours vs plane via Paris or Barcelona means bus would seem to be best choice unless direct flights Bilbao-Barcelona.