We are driving from Fussen to Bormio on Sunday 3rd June in our own car and, if the weather permits, we want to drive over one of the passes. We had intended to drive the Stelvio but then I realised that the Saturday before is Republic Day and I am now wondering if it will be too busy as it is a holiday weekend. Is it better to take the Umbrial Pass to avoid heavy traffic or is it likely to be just as busy?

La Festa della Repubblica is on June 2nd.
I cannot see why there should be more road traffic in the Provinces of Sondrio and Bolzano than on ordinary weekends.
The road from Stelvio to Bormio goes via Umbrail anyway.
You will just have the choice between
Reschen Pass - Mals - Glurns (smallest walled cirty of the Alps) - Mustair monastery (Unesco site) - Umbrail - Bormio
and
Reschen Pass - Mals - Spondinig - Trafoi - Stelvio - Umbrail - Bormio.

Here you have the events in the Stelvio: https://www.bormio.eu/en/events/2018-06-03/?lang=en
There seems to be some biking event on the 3rd so maybe you could try to avoid it.

Ahh! How did I miss GRANFONDO STELVIO SANTINI. Thanks so much Ticino.
I have emailed the Bormio tourist office to find out how long the roads are closed as it looks like both routes will be out and we may have to use the Munt La Schera tunnel instead.
Neckervd - thanks for your comment. If we have to use the tunnel is there much difference in routes?

The itinerary via Munt la Schera is completely different between Pfunds and Bormio:
Pfunds - Vinadi - Martina - Scuol-Tarasp - Zernez - Punt la Drossa - Munt la Schera Tunnel - Livigno - Foscagno Pass - Valdidentro - Bormio.