I am not an interest rate product person (the likely setting for your question), but I will take a shot at it anyway.
The Black model is a one parameter model. Once you calibrate it to a given option price with a strike and term, you have one parameter: the implied volatility. Then, what? That one parameter, by itself, typically doesn't help much with other strikes and terms. Of course, you can interpolate the IV surface. More on that below.
Now suppose you have a `good' multi-parameter model. By `good', I mean once you've calibrated the parameters, you can use the model -- with those same or perhaps similar parameters -- for various strikes, terms, dates, products, etc. You can also translate those model prices to Black implied volatilities.
Whether or not some shifted SABR model is 'good' I leave to the fixed income folks.
So, yes, you can use any model by calibrating its parameters, generating prices, and translating model prices into Black implied volatilities. But there is no guarantee this procedure will be helpful. There are lots of issues. Are the parameters stable, model prices close to the market, etc? Is the procedure an improvement to simply interpolating/extrapolating Black implied vols -- without introducing the second model?
Be skeptical. It's always possible people are doing something, in a particular application, just because other people are doing it. There may be some more-or-less `model-free' way of accomplishing the same goal, such as via the Breeden-Litzenberger relation.