Technically, you are correct that if your road association is not a "statutory" one set up under 23 MRSA 3101, you are not bound by that series of statutes; however, the reasons that statute requires members to be allowed to vote still exist. If you think back to the American Revolution, one of the reasons we split off from England was because we became frustrated by "taxation without representation." Our country is founded on (among other things) the belief that a person should have some say in how his money is spent.
On the other hand, the U.S. Constitution prohibits states from making or enforcing laws that impair the obligation of contracts. If you have a non-statutory association, its bylaws would be a sort of contract between the members. Members who have agreed to what is in the bylaws should be bound by them. Then the question remains as to whether the state can prohibit a statutory road association from having a contract that allows voting rights to be withheld. I am NOT an attorney, but I'd have to say there is room for debate here. The Constitution also prohibits "involuntary servitude," i.e. being forced to work without pay, but that leaves the door open for "voluntary servitude" - choosing of one's own free will to work without pay. Similarly, I don't see why a person couldn't agree in a contract to give up their voting rights if they refuse to pay. Bottom line - do what seems to be morally right.
There probably is little danger that non-paying members would out-vote paying members, so denying them their vote is probably more a matter of trying to give them incentive for paying their dues. I do know of one non-statutory association that had loss of voting rights in their bylaws and was forced to use it because they had more non-paying members than paying members. The association ultimately folded for lack of funds. They then formed a statutory road association, and have gained several paying members each year since then, thanks in part to people seeing that the road is actually improving, and in part to the enforcement tools available to a statutory association. Their statutory bylaws were modeled on the old guidelines and do include loss of voting rights for unpaid members, but they have never had to invoke that rule since going statutory, because non-paying members never show up to vote anyway. They are planning to vote to correct their bylaws to eliminate this provision at their next annual meeting.
I have one other concern about your situation. You have said that the DEP, the county, and the town have designated your road as a "private way." I would be extremely careful about the use of that term, as it has multiple conflicting definitions in Maine law. Do they mean that it is a road that is privately owned and maintained, and over which the owner(s) can restrict passage, as defined in 29-A 101 paragraph 58? If so, to avoid confusion I would use the term "private ROAD."
On the other hand, 23 MRSA 3021 defines a private way as a public easement. Under title 23, the term "private way" could mean that it's a road that used to be a public town or county way, but public maintenance has been either officially discontinued under 23 MRSA 3026-A (or its predecessor, 3026,) or abandoned under 23 MRSA 3028. Either of these actions make the road a "public easement," but earlier versions of section 3026 used the term "private way." In either case, under section 3021 the road remains open to unrestricted public use by foot or motor vehicle in spite of the fact that there is no longer any public maintenance. That gets into another whole can of worms, due to questions of whether it's right to force private individuals to pay to maintain a public road, over which they cannot restrict public traffic. (Yet another example of taxation without representation, and involuntary servitude!) So I hope your "private way" is actually a "private road" - one that was privately laid out and built, and which has never been accepted as a public road.