Wouldn't those lobbying regulations (blocking federal employees from lobbying) violate the first amendment? Wouldn't that deny an entire political class of people from being able to participate in the political process?

I assume any such law wouldn't block their speech, just their ability to receive money in exchange for that service

Except under Citizens United money is speech. How is this supposed to not conflict with that decision, unless they mean to clarify Citizens United is that only corporate personhood can use money as speech?

Money is speech

obviously I am not a supreme court justice, but its not limiting speech or expression. It is explicitly preventing what many people consider to be a gross conflict of interest. It's post-fact in a way, but its still messed up.