Yes you can

The rules for wildshape clearly state that you retain any benefits of features from your race if the new form is capable of using them. Since the Relentless Endurance feature does not implicitly or explicitly require any particular body parts (hands, for example) I see no reason why it should not be retained.

The rule on reverting says you automatically revert if you hit 0 hit points. Since your Relentless Endurance allows you to drop to 1 hp instead of dropping to 0, you never actually drop to zero and thus never revert.

... and there is no carry-over damage

The rule on carry-over damage says:

if you revert as a result of dropping to 0 hit points, any excess damage carries over to your normal form

Since you did not revert or drop to 0 hit points, this rule does not apply.

There are no rules that state that a wild shaped druid is not able to take or benefit from a short rest. There is a clearly defined list of what they cannot do in the PHB.

As there is no requirement to maintain concentration or focus on the shape, and because transforming does not break concentration on existing spells, a druid is able to meet the requirements for a short rest.

PHB Page 67
You retain the benefit of any features from your class, race or other source, and can use them if the new form is physically capable of doing so.

PHB Page 186
A short rest is a period of downtime, at least 1 hour long, where the character does nothing more strenuous than eating, drinking, reading or tending to wounds.

Unless your form is described as a never stopping ball of energy, most forms are capable of resting, and therefore are capable of regaining uses of wild shape.