Assuming some number of shareholders believe that the bankruptcy process is inadequately compensating them (i.e. by setting the company valuation too low and thus giving creditors too much equity in a new entity), is there a possibility of exercising appraisal rights as there would be in a merger?
Shareholders rights are eliminated in their entirety in a Chapter 11 bankruptcy unless all claims of the bankrupt company are paid in full under the bankruptcy plan adopted, in which case the shareholders simply keep their shares when the company emerges debt free from bankruptcy (this almost never happens).
The amount that shareholders receive is based solely upon what is left over after all creditors are paid in full. But usually, in a Chapter 11 bankruptcy in which the plan succeeds, because the debts to be paid under the plan have been paid, no distribution is made to shareholders anyway, even if the shareholders continue to have shareholder rights because all other creditors have been paid in full.
So, no, shareholders do not have any possibility of exercise appraisal rights as they would in a merger.