Can I use the Swiss Travel Pass to travel to Innsbruck, Austria from Bern, or is it strictly in-Switzerland?
It is only valid to Buchs SG, the last stop in Switzerland. You can put in that you have a "GA" card (that is a pass that includes a valid Swiss Travel Pass) when it askes for your discount, and see if you save any.
If you add the GA discount you will get a reduction on the normal price ticket. You will probably get a better price if you just book Buchs SG to Innsbruck with the Austrian Railways (www.oebb.at).