After our stay in Arles I planned on taking a train to Brugges, Belgium (from Avignon) on May 19th. Even until a couple of days ago it looked like there were many trains running that route but now that I am going in to book it says that no trains are available. Does anyone know why this might be? Could it all be sold out 3 weeks in advance? Are there other alternatives to get to Bruges from Provence or would I just have to trek back to Paris and go to Bruges from there?
The problem is likely because this is a cross border trip, and SNCF sometimes doesn't play nicely with these. You could book a TGV back to Paris on the SNCF site, then an onward trip to Bruges on the Belgian rail site. This would require a change of stations in Paris. Or try booking through the Trainline website. This is a 3rd party site that will charge a booking fee, but may be easier.
You can try the international website of the Belgian national carrier NMBS/SNCB for booking without a 3rd party. Some trains are indeed fully booked, but still enough seats available.
The 1147 looks to be the best train as that avoids a cross Paris transfer.
That train is also available to buy as far as Brussels on SNCF connect. Then buy a local Brussels to Brugge (Bruges) ticket at Brussels.
It's quite hard to get to Bruges without changing in Brussels. So I suggest you consider SNCF/Oui to Brussels. From there, there are at least five unreserved local trains an hour in the daytime. Those local trains don't always have long-distance luggage racks.
I see these on the Belgian rail site, https://www.b-europe.com/EN.