Is it permissible for a common Muslim who trusts two different authorities to usually follow one of them, then sometimes follow the other in pursuit of what is easier, and is this considered an impermissible pursuit of dispensations, and must he make up the prayers he performed in this manner?

If both parties are equal in knowledge and piety, then the inquirer (mustafti) may follow whichever of them he wishes. Scholars have unanimously agreed on the permissibility for an ordinary person to follow any of the scholars whose knowledge and piety are trusted, without seeking out dispensations. This is the view of the majority of Hanbali scholars, who argued that an ordinary person has a choice in following any mujtahid in subsidiary matters, and is not obligated to exert effort in determining the best mujtahid. Based on this, there is no blame upon you for following either of the two parties. And if one of them issues a fatwa to you that you are not obligated to make up the prayer, then you may adopt their opinion.