The question itself makes no sense. One does not install the AUR, but rather individual packages from the AUR and those would expect a distro compatible with them to be present in DB. Though once again what do you mean by whole distro. Arch without cruft is barely anything.

What do you mean by "entire" distro? The distro itself doesn't download the "entire distro", it's made up of packages, like almost every distro is. You're only installing the base system, the packages are where the rest of the downloads come from. What "parts" of the distro's base system don't you want? The dockers are already pretty stripped down and minimalist by default, I'm not sure what else you could cut out to still have a functioning OS?

Sure, just install paru or yay and proceed as normal.

This might be an xy problem.

What are you trying to solve?

If you're just trying to have access to an community package manager.

One of these might be more useful.

https://flathub.org/en/setup

https://nix.dev/manual/nix/2.34/quick-start

If you specifically must have the AUR. I suppose you could install arch in a container like you suggested.

No

What do you mean by "install aur"?

What are you trying to do, specifically? And why? AUR is a collection of third party package build scripts to install software. You don't download the entire thing on any system.

If by install AUR inside distrobox, you mean run a distrobox that has access to the AUR? Then, yeah, you'd need the Arch container for distrobox and either install pamac or yay