Does a Static $E$-field Increase the Gauge Invariant Vector Potential Without Bound?

This gauge invariant vector potential increases without bound as long as there exists a static electric field. Indeed, even when the electric field is removed, there appears to be no mechanism by which the gauge invariant vector potential disappears.

Static electric field has zero transversal component; entire field is longitudinal.

The unbounded increase of vector potential would occur if you used the gauge where the static electric field is given as time derivative of vector potential.