This is a tricky one for sure, but it still takes a backseat to Sanctuary S Rank or 3hp Best Ending.
Due to the 8min time, you do have some leeway in your run for little mistakes, but not too much. From my experience, and from watching fast videos, I've put together a list of weapons that are effective for killing the bosses quickly, and have a video demonstrating a run with quite a few minor mistakes that is still sub 8.
Balrog - Polar Star (If you're fast enough you can kill him before he reaches you)
Toroko - Use all missiles then unload polar star into him
Balrog - Unloading missiles will kill him
Balfrog - Run into him (with polar star to not delevel) to get to his other side quickly and
unload missiles into his mouth. Possible for him to die in 3 rounds of missiles,
otherwise switch to fireball 3
Curly - Fireball 3
Omega - Fireball 3
Rabid Toroko - Fireball him as much as possible, missiles up when he jumps over you
Puu Black - Unload all missiles into him; possible to kill him before he jumps
Monster X - Blade 2 worked best for me; try to kill all the green things in one go
Balrog - Blade 2; when he runs at you, run away while shooting and you can hit him quickly.
I am pretty bad at pulling this off though.
Core - Super Missiles and blade 2 if you run out
Iron Head - With maxed super missiles you can kill him before he goes offscreen; otherwise
when he falls back but that'll be an extra 5 seconds or so.
Talk to the guy before the twins and decline his trade for the snake; this allows you to get the spur which feels absolutely necessary for the speedrun.
Twins - Charged spur then missiles into stunned dragons or just use missiles caerefully.
Ma Pignon - Level 2 charged spur is fastest; if it takes too long that it jumps up, you can get
a full charge into him and that should be enough at that point
Red Ogre - Easy destruction with super missiles
Misery - Blade 2 was my go-to, but some prefer using the spur; don't stray too far from the
center since she will teleport randomly every 90hp or so
Doctor/Muscle Doctor - Super missiles will destroy him
Undead Core - Blade 2 on the non-rabbit minion, then blade 2 on the core until it stops
taking damage. Then just keeps charging your spur and firing when it opens
up
Heavy Press - Super Missiles, but make sure you max them beforehand because you will
want super missiles maxed for Ballos later as you'll take damage from the
press attacks.
Ballos 1st Form - Blade 2, go near him and fire away upwards while dodging.
Ballos 2nd Form - Blade 2, but turn around so that you get the fire rate of blade 2 but with
the power of curly's nemesis
Ballos 3rd Form - Mix the nemesis and uncharged spur; if you hit bats, dodge the
experience if you have the nemesis selected, or switch to spur and pick it
up while firing away.
Ballos 4th Form - Super Missiles while standing in his eye (I made a big mistake here using
them in part 2, and failing the boost a few times)
This is how I tried to play for most of my runs, with some minor forgetful mistakes (Ballos almost ruined my sub 8 because of poor judgment), but keep at it and you'll have one of the hardest achievements in the game! Watch the video for a more visual demo of the boss descriptions.