It's possible. BAT actions are a very consistent standard. In fact, if you had a barrel finished for your SV action, I could use it just the same on mine. It would need similar tightness, and I would check the coned breach clearance as actually clear and no more than 5thou.
If you purchased all barrels from the same finisher, who made a bunch with the same action/go/no-go gauges, there may be no difference with barrel changes. I witnessed this through 3 barrels with a BAT RS action. Same brass & die settings, same load, zero change across barrel changes. Says a lot about the barrel finisher as well.
Other than Savage, I don't think this would work out from a common factory action.
I love my new BAT VR action. They did some extra work to remove some material for weight reduction. I just got the barreled action back from my smith. Waiting on the stock to get finished. It should be a very sweet shooter. Very smooth. Im gonna start barrel break in soon.