However, he is giving more power to the relatively few people who might vote in his "poll" than the same number of people in another district who have no direct control over the vote of their representative. This is the "equal protection" issue.
That's not an equal protection issue. The unenforceable pledge of an elected official to vote X because his constituents say so does not rise to the level of state action to be protected under the equal protection clause. Just like a broken campaign promise can't be prosecuted as election fraud. He has a duty to represent the people of his district and how he chooses to do that is his business. The law says that X district shall elect Y members who shall represent them faithfully in Z body. Additionally, there is nothing stopping the other members from doing the same thing, thus they are protected equally.
Even if there WAS an equal protection clause violation, it'd be subject to the rational basis test. It'd be constitutional if there's a rational basis for it and it serves a legitimate government interest... which this does.
Again, even if there WAS an equal protection clause violation to be found, there is no equitable court-enforceable remedy for it, therefore it's moot. You cannot enjoin him from serving. Nor could you really enjoin him from proceeding as planned (more of a logistical limitation). The Vermont Senate has control over the conduct of its members and they could expel him, if they saw fit (not definitive as I haven't read the rules or the Vermont Constitution on this).
That, and it is a unilateral change to the design of the system that applies only to some of the people who are operating under that system. Every other district will have a representative, this one district will have a proxy system. Does this not result in less protection for minorities from the "tyranny of the majority" in his district than those in other districts?
It's a not a "change" and it wouldn't be unilateral. It would be a voter-mandated philosophy. Protection for minorities from the tyranny of the majority is not a Constitutional guarantee nor protected under the equal protection clause. Otherwise, actions taken during pro forma sessions of the Senate when there are only one or two members around would not be valid, for the protection of the majority against the minority. Elections where a plurality instead of a majority were obtained would also not be valid, for the protection of the majority over the minority.
Additionally, the Vermont Senate has multi-member at-large districts and he would be 1 of 3 members from that district. The people of the district are still adequately protected.
Even if 100% of the lobbyists visiting one Senator's office want him to vote a certain way, there is no guarantee he will. This guy is promising his constituents that he will vote the way the majority of them want him to on everything. That's significantly different from just being a lobbyist.
The percentage of persons participating in his experiment would never be 100% of his constituency, thus, they are just like lobbyists. Elected officials make voting promises to lobbyists routinely.