First, it'd be illegal to do in secret. Second and I suppose more significantly, it would not offer a military advantage over actually having them serve their intended purpose as defensive weapons (which would be useful in any hypothetical nuclear exchange, including a US first strike) and then moving a ballistic missile submarine or two to the Barents sea, or equipping some of the many attack submarines operating in those waters with nuclear-tipped cruise missiles, or some combination of both.
There isn't any need for secrecy. An anti-ballistic missile is also a missile. So they can theoretically be designed to be both defensive and offensive. That's what the indians did - converted one of their offensive missile to also be a defensive missile that can shoot down other missiles (I am not claiming that the Indians are using the same missile both as an offensive arsenal and in their missile defence shield, but just that it may be possible in the future for them. Both Russia and the US are well ahead of India in missile R&D, and it may not be far fetched to think that they may have already developed configurable missiles that could be used for both offence and defence.)
The Russian hypothesis we're discussing here is that these are not defensive weapons, they are secretly offensive weapons.
Strictly speaking, there would be a legal need for secrecy until we're fully withdrawn from the INF treaty. You'd also have to develop the nuclear warhead to fly on the interceptors in secret. They'd have to be awfully lightweight, since these interceptors aren't designed to carry a nuclear warhead. (a quick look at wiki indicates the RIM-161 uses a kinetic energy warhead, which is a little hint at how silly it is to even be having this discussion) So that's a weapons program you'd have to fund in secret, as well as the funding needed to adapt all those weapons systems to a completely different purpose and test them.
You'd have to keep the whole thing secret from those among our NATO allies who would never approve such a thing. Great idea.
None of this would give the United States any first strike capabilities it doesn't already have thanks to its navy. I'm trying to be polite but this is all very silly. It's just not what those systems are there for.
First, it'd be illegal to do in secret. Second and I suppose more significantly, it would not offer a military advantage over actually having them serve their intended purpose as defensive weapons (which would be useful in any hypothetical nuclear exchange, including a US first strike) and then moving a ballistic missile submarine or two to the Barents sea, or equipping some of the many attack submarines operating in those waters with nuclear-tipped cruise missiles, or some combination of both.