My money's on ZDoom.
Really the only difference is the perspective at which it's played if the mechanics are the same. Also I think it's a good idea to experiment even if the combination doesn't sound good on paper.
Gimme your money: Enhanced Doom Gaming Engine. a.k.a. EDGE Engine ^_^
It bugs me (not much, but still) when people don't have something to prove they're working on a project but post about it anyway.
The worst part for me is the creation of the game art.
I'm a terrible artists, so I need to rely on free art around internet.
That is why all monsters anr zombieman, attacks are player_missile and so on. They are just placeholders. And of course, I'm not going to use realm667 monsters because I want enemies to be anime style, even if having only a single face.
Take a look at Demonsteele - it's a (Z)Doom mod with an emphasis on sword+gun combat and enemy bullets. It's probably not really what you're trying to achieve, especially since you won't really get a bullet hell experience without playing slaughter maps with it and the fact that a sword is the main weapon there, but it might still give you some inspiration about what to do.
It won't be Doom anymore, not even play like it at all.
There won't be a map that you can explore like in your video example.
The map is basically a corridor without visible walls, ceiling and floor. There are "control" monsters that, once killed calls the appropriate script which spawns the next wave of monsters.
But I'm also considering doing the wave spawing time-dependant rathed than monster killing dependant.
It is not still implemented, but I plan to do 3 player classes, pretty much like Touhou 2. There won't be BFG, chainsun or plasma rifle, only Touhou style attacks.
Unimplemented also, the bonus and power system. Killing monsters may spawn points or weapon power bonus.
