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.
A druid can only wildshape on his turn. So, while he can apply unlimited hitpoints to his form by wildshaping every turn, he cannot prevent you from dealing damage when it isn't his turn. If the druid is knocked out of his form (reduced to 0 hp), the first thing that happens is the excess damage carries over. But now he's essentially vulnerable until his turn. That's where the real damage comes in. You can think of wildshape as a regenerative shield of hit points.
Party Versus NPC Archdruids
There are a multitude of ways to "beat" the druid, just like any other encounter. Use your wits, bargain, stealth, combat, etc etc. Those tools are ALWAYS available. Your DM should be ready for you to avoid combat if you wish.
Party Versus PC Archdruids
If you're talking about PC vs PC combat, there are also a plethora of spells that can assist in killing the druid. For instance, if the druid has less than 100 HP in wildshape OR not, a Warlock can cast Power Word: Kill, and the druid simply dies. No save, just dead. That's just one spell. There are tons of others. A Druid can't wildshape if they're incapacitated. They, again, also can't wildshape unless it is their turn, so plan your heavy attacks accordingly. When a druid is knocked out of form, they are vulnerable, so maximize your damage in that time or make that vulnerability last longer with controlling spells. Any damage that exceeds a druid's wildshape form is applied to the druids regular form. Druids are very good at having health, so find a way to attack them in ways where health isnt the primary objective. Use conditions, spell effects, and any means to hinder the druid and he/she shouldn't be that tough of a fight.
DnD 5e rewards specialization. An Archdruid specializes in wildshape to effectively gain temporary hitpoints among other combat abilities. It's the whole point of a wildshaping druid. Being a brown bear is cool, but when you can wildshape into one and get its free hitpoints, it becomes a big deal. That's largely what wildshaping druids are meant to do, so it only makes sense that it gets ridiculously good at it at level 20. All classes have a 20th level ability they get that helps them be even more specialized. It's a reward for forgoing multiclassing. Therefore, it should feel hard to deal damage to an Archdruid. But it's not impossible.
To answer the first question: No. It would not carry over due to how Wild Shape works.
The reason: The only reason damage carries over is because you automatically revert when your HP is reduced to 0 in beast form. This means that damage is dealt after you revert technically.
Therefore the carry over of damage between forms is delayed damage to your caster form until you revert. No effect of healing causes you to revert to receive the extra restoration of hitpoints lost.