Essentially you're going to want:

• Dhewm 3 Source Port (dhewm3.org) [Vanilla Friendly],
  
• The Lost Mission for Doom 3 (2004) (ModDB) [Vanilla Friendly],
  
• Doom 3 Essential HD Pack (ModDB) [Vanilla Friendly],
  
• Doom 3 Rebalance (Nexus Mods[www.nexusmods.com]) [Gameplay Changes].

Final Product:

First download the source port Dhewm 3 from https://dhewm3.org/

It should take you to the latest version on GitHub, scroll down and find "dhewm3-1.5.3_win32.zip" for Windows.

Extract the files to your Doom 3 directory, results should look like this:


You will now use dhewm3.exe to run Doom 3.

NOTE: Anti-aliasing can cause a lot of performance drop, I leave it disabled and use ReShade instead.

You can now play The Lost Mission in the original 2004 Doom 3. It is compatible with the Dhewm 3 source port and the mods discussed here.

Download The Lost Mission here:
https://www.moddb.com/mods/the-lost-mission

The Doom 3 Essential HD Pack can be downloaded here:

https://www.moddb.com/mods/d3hdp-doom-3-essential-hd-pack

It also comes with many optional add-ons.

Optional mod pack that tries to re-imagine the Doom 3 survival horror experience.

Install this mod pack by extracting its contents to your Doom 3 directory.

Mod pack:
https://www.nexusmods.com/doom3/mods/1684

Contents / Credits:
Full detail on Nexus Mods page.