I don't see how there could be an argument for protected speech here. The constitution protects your ability to petition the government, not the government's ability to petition you.
If someone is campaigning to be in government, I guess they don't technically fall under the "government" category yet...
I don’t know if the claim is true, but if it is, it’s likely an intentional loophole because the people writing the laws would want to exempt themselves.
A candidate is not acting as a representative of the government. In fact, the Hatch Act generally makes it illegal to mix the two.
https://en.wikipedia.org/wiki/Hatch_Act