It is possible, but not in easy way ;)
Problem is, you would have to re-did all the menu buttons and other elements (some can be copied from RoE files, but not all). Otherwise, you'd get ugly "hybrid", like this.

bad example 1
bad example 2
bad example 3

That said, it can be done, if you have some spare time on your hands. At this very moment, I don't have much :)

I took them from RoE disk. In complete edition there were probably replaced by SoD counterparts.

Yep, it is :)
I think I'm gonna tinker a bit more with it at home, but tomorrow ;)

Okay, I think I more or less manged to do it. And I tried to keep it easy to use ;)
Here is the download link: (clicky)

Now what to do with it:
- unpack archive to your HoMM3\DATA folder;
- there, inside 'ArchangelBg' folder you have two .cmd files
- run background_archangel.cmd (as administrator) to get RoE background in-game
- run background_restore.cmd (as administrator) to restore backup and original HoMM3 Complete background

Now, couple of notes:

* Buttons aren't perfect, especially campaigns :( I know, but I don't have original font (it's commercial). Soo... just don't look at them too long ;)
* Don't change folder structure, it should look like <your_homm3_directory>\DATA\ArchangelBg\(mod_files)
* Everything was possible thanks to GrayFace's various tools

and kinda obvious but I have to state it:
These files are no way related to nor supported by GOG.com. It's merely a bit of personal fun of mine :)

No idea, I don't use HD mod, but I guess it changes some resources on its own.
I'll try to find some time to check it tomorrow.

It's not a matter of scripts, I'd have to redid all the buttons again :O
I'll take a look but no promises.

changing loading screen is even easier ;)

It was tested only on vanilla install, not with HD mod.

Edit: oh, and you can edit .bat with notepad, simply add 'pause' above 'exit', save and run. Screen won't close without your input.

It can work with HD, but without animated background.
edit both .cmd files and remove any lines with 'video.vid' (or simply add 'REM' at the beginning of each line).

Unfortunately I never backed up CRW files, but is someone manages to find them again I can put them on GDrive and that shouldn't expire like previous links.