Hello I am a solicitor with 20 years experience. I will try to answer this. They have no legal right to leave their bins on your lsnf even with the right of way.
However I do not think that getting an injunction is a practical option. The harm you are suffering is unlikely to be viewed as serious enough to merit an emergency injunction and you would therefore need to issue proceedings in the court and wait until the final trial of the issue to try and seek an injunction which might well be granted then. It would be very expensive to do this. But even if you did it would be hard to enforce.
I think your family solicitor is right and that isn't something to pour money at in legal costs.