Custom packages will almost always require a rebuild after a major update - especially on stable branch.
You should be aware that you need to be on Manjaro unstable branch to be able to use AUR.
Execute the following command - it should provide a similar response - that is - if you want to run custom packages using AUR
$ sudo pacman-mirrors -G
unstable
ungoogled-chromium-bin run fine on unstable branch
Do not replace system libraries with custom packages from AUR - not ever…
Refreshing your mirror list will not provide any changes that affects building custom packages from AUR.
Assuming there is no packaging issues - the following is the recommended approach
sudo pacman-mirrors -aS unstable
sudo pacman -Syu base-devel --needed
git clone https://aur.archlinux.org/ungoogled-chromium
cd ungoogled-chromium
makepkg -iscC
And yes - relying on a package like ungoogled-chromium is a pain - the only way you can remain stable in such case is to create a local (portable) mirror