When does Lavender bloom?? and Arles question...

It can be difficult to answer exactly when the lavender blooms in Provence due to weather conditions for any particular year. This website gives you a rough idea when it's likely to peak in the different areas of Provence. The Abbaye de Senanque (I assume that's what you mean by the Lavender Abbey) is near Gordes, about an hour from St Remy.

Yes, Arles is on the way driving from St Remy to Barcelona so worth a couple hour stop if your kids are interested in it.

I passed by Sénanque abbey 10 days ago and the lavender only had green buds at the tip of the stems, so I would not expect a full bloom for another 10 days or so, that is to say mid-June which is typical for the area.
If you do explore some villages in the Luberon, you will drive by lavender fields. There are a few between Gordes and Roussillon, and there are many along the road between Bonnieux and Saignon (D232) on the Claparèdes plateau. The latter are really stunning, with a great view of Mont Ventoux in the background.

As for Arles: not silly as long as it is not market day! It is not out of the way to Barcelona. With a luggage-full car, use "parking du Centre" car park which is secure indoors and just a short walk to the theater and amphitheater.
On a market day (Wed or Sat), I would not want to deal with Arles. An alternative could be Aigues-Mortes, a beautiful walled city in the western reaches of the Camargue, easily accessed by fast roads (you rejoin the main highway to Barcelona in Montpellier).