We will be traveling from Amsterdam to Avignon on May 1st (Labor Day Holiday). Will we have any trouble securing train travel?
I would suspect that the day before would be the mad rush, just like the Wednesday before US Thanksgiving is the crazy day in the States. May 1 falls on a Thursday. You might start shopping after the New Year to make sure you get the times you want at the best prices.
This year we traveled by train on May 1st from Avignon to Caen through Paris with no difficulties. The only non train problem was finding a car rental agency that was open in Normandy. We had to settle for getting off in Caen as the Hertz office was the only one open during our arrival time.
May 1 is not a holiday in Holland. It is in France but transport keeps running though maybe on a reduced schedule. Labour Day in Europe is somewhat different from North America. In France there are likely to be union and workers parades in the major cities but that is about it.