Once the snapper becomes the snapper, by rule, he may not move to another position. Except in a true scrimmage kick formation (which a swinging gate is NOT), a player does not become the snapper until he takes his position at the ball and touches the ball or simulates touching the ball by placing his hand(s) at or below his knees. So, in your scenario, he never became the snapper, by rule. Since he wasn't the snapper, he is free to change positions.
Note that, in a true scrimmage kick formation (as of 2026), a player becomes a snapper when he simply takes his position on the line at the ball. There is no requirement for him to address the ball for him to be "locked in" to his position in the "2X2". We should never confuse a swinging gate with a true SKF, but we must know the difference in the two formations, since it affects what Team A is allowed/not allowed to do with respect to player numbering, and what Team B is allowed/not allowed to do with respect to the alignment across from the snapper, and blocking of the snapper.
Good discussion.