Belt contents are immutable; the belt is strictly SSA. However, although you cannot move a value on the belt, you can create a new value that is the same as an existing value, and your example of adding zero would do that. OR with zero would likely consume marginally less power than add with zero, especially if the operand you want to save is a vector.
So there’s a better way: the rescue op. This does the same belt normalization that control flow operations do, except without actually transferring to a new address. Rescue comes in two forms. One specifies the belt positions to move to the front using an argument list. This form can arbitrarily reorder the rescued operands, and can also duplicate operands; in this is acts exactly like (and uses the same hardware as) the call and retn operations. However, the position-list argument uses one morsel per position of encoding space.
The other form of rescue specifies the desired operands using a bitmap of the positions, which are moved to the front in numeric order and dups are not possible. The underlying implementation and effect is exactly the same, but the bitmap is a more compact encoding if you rescue more than two operands.
Because rescue is done entirely in the name-mapping logic it has zero latency, just like the equivalent functionality in branches and calls. It also does not require a trip through an ALU so it uses less power than e.g. adding zero, especially as the name-mapping is updated every cycle anyway. And although the belt effect is the same as that of a branch to the locally following instruction, there is not actual control flow and so the transfer-prediction hardware (with its power overhead and risk of prediction or icache miss) doesn’t get involved.