Kill a few Minecraft dogs in front of him, then whisper “you’re next if you don’t pipe down” at him.

Likely, he’s been rewarded with attention when he howls while someone is using a computer. Reconditioning the antisocial behavior entails rewarding him for the social behavior you prefer.

He’s a beagle, so he’s stubborn and not willing to negotiate

Well have you tried offering him a turn? You've been hogging the controller, let him try.

@LaunchesKayaks it's probably a seating issue.

Get one of these chairs

Try telling him this: Woof! Woof woof! Bark woof grrrrr woof! Bark bark rrrrrrrr bark raa raa raa woof.

woof.

You've wrongly linked the behaviour with an action. Beagles yell regardless of what you do if you're not paying them attention.

You can't. Play Terraria instead.

My dog absolutely hates certain noises. Anything excessively high pitched especially.

Have you tried turning the sound off, or is it you being on the computer in general rather than playing Minecraft?

Do you play any other games? If not, it may be the higher load than normal causing the fans to spin, and maybe that's messing with his ears.

The only other thing I can think of, besides the dog simply seeking attention, are your actions while you play Minecraft. Whether you make noises or yell, fidget, gasp, move in your chair a lot (squeeky wheels?) etc.

Frozen treat stuffed inside a Kong toy. You have at least 30 minutes. Lol

Does the dog get upset when you have headphones on? If it's a beagle the sounds and or music may be messing with it's ears.

Have you checked to make sure he is not your human dad?

Invent Minecraft for Dogs.

Why play Minecraft when you could be playing with a real dog?

Be considerate. https://audiochamps.com/does-loud-music-hurt-dogs-ears/

Feed him some rotten flesh

Yell back

Get him a BeagleBone Black!

Play Minetest and see how he reacts.

Tame a wolf and then kill it in front of him. Maintain eye contact while you do it. He'll get the message.