First question - if the beaver activity damages the road, will that affect access to the property that doesn't want to allow you to deal with it?  If so, that gives you a bargaining chip.  If not - oh well, it was an idea.

Maybe get in touch with Inland Fisheries and see if they have any suggestions?  How wide a right of way does the association have rights to?  Would a beaver deceiver be within the right of way bounds, and therefore within the association's rights?

Have you tried contacting Inland Fisheries to see if they have any suggestions?

Another possible source of help is USDA Natural Resources Conservation Service (NRCS).  They assist private landowners with issues with stream crossings.   I'll attach a brochure.

I've been trying to think if there are any statutes that might help you, and the closest I can come is the following: 

Title 17 §2802. Miscellaneous nuisances

The erection, continuance or use of any building or place for the exercise of a trade, employment or manufacture that, by noxious exhalations, offensive smells or other annoyances, becomes injurious and dangerous to the health, comfort or property of individuals or of the public; causing or permitting abandoned wells or tin mining shafts to remain unfilled or uncovered to the injury or prejudice of others; causing or suffering any offal, filth or noisome substance to collect or to remain in any place to the prejudice of others; obstructing or impeding, without legal authority, the passage of any navigable river, harbor or collection of water; corrupting or rendering unwholesome or impure the water of a river, stream, pond or aquifer; imprudent operation of a watercraft as defined in Title 12, section 13068-A, subsection 8; unlawfully diverting the water of a river, stream, pond or aquifer from its natural course or state to the injury or prejudice of others; and the obstructing or encumbering by fences, buildings or otherwise of highways, private ways, streets, alleys, commons, common landing places or burying grounds are nuisances within the limitations and exceptions mentioned. Any places where one or more old, discarded, worn-out or junked motor vehicles as defined in Title 29-A, section 101, subsection 42, or parts thereof, are gathered together, kept, deposited or allowed to accumulate, in such manner or in such location or situation either within or without the limits of any highway, as to be unsightly, detracting from the natural scenery or injurious to the comfort and happiness of individuals and the public, and injurious to property rights, are public nuisances.

https://legislature.maine.gov/legis/statutes/17/title17sec2802.html 

It's not an exact match for your situation, but the principle similar, that is, a person should not be able to do something that damages other people's access to their property.

I wonder what made the person object to the beaver deceiver, and/or if they were the ones who removed it.  Could it be replaced without going onto the person's land? 

I'm thinking it might be helpful to get a professional mediator involved to contact the person and listen to their concerns, and find out what caused them to have their current objections.  Are they nature lovers who don't want the beavers disturbed?  Or do they have a grudge against someone who needs the road for access?  Did someone trespass on their land?  There could be any number of things motivating them, and if you can find out what has them upset, perhaps something could be done to address their concerns first, and then resolve the issues with the road.  Such conversations can be delicate, which is why I would recommend getting a professional mediator rather than trying to do it yourself.  The Maine Association of Mediators has people with the skills to handle it.

Unfortunately I cannot tell you how much Mediation would cost.  It used to be possible to get a Mediator from Family and Community Mediation (FCM) at nominal cost, but unfortunately that organization no longer exists.  But many of the Mediators who had been involved in that effort are members of the Maine Association of Mediators, and might be willing to negotiate an affordable rate - I don't know that, I'm just speculating.  I believe most charge less per hour than an attorney, although the rate probably varies depending on who you hire.  Some offer a free preliminary interview.  The big saving is that your case will probably be settled a LOT sooner, both due to a more direct process, (i.e. no attorney hours spent researching related cases and preparing court filings,) and due to avoiding the court backlog.  

To find out more, go to the Maine Association of Mediators website, here: https://www.mainemediators.org/