OK, first off, while Bowser does outclass Eggman in a lot of ways, they are also bringing their respective armies into battle as well. Bowser will have his Koopa Troop and Eggman will have his Eggman Empire. In the Koopa Troop, there are Goombas, Koopa Troopas, Shy Guys, Dry Bones, Piranha Plants, Chain Chomps, Hammer Bros, Paragoombas, Parakoopas, Boos, Cheep-Cheeps, and Bloopers. The Eggman Empire has Eggpawns, Eggfighters, Eggrobos, Moto Bugs, Caterkillers, Swatbots, Slicers, Buzzbombers, Robo Chomps, and so much more. However, while Eggman's army is stronger, faster, and tougher than Bowser's minions, the Koopa Troop always work in numbers, and the Eggman Empire doesn't usually work as a team. But some of the stronger members like the Koopalings, King Boo, King Bob-Bomb, Kamek, Bowser Jr., Silver Sonic, Mecha Sonic, Mecha Knuckles, Sage, and Metal Sonic will be tougher to bring down due to their better durability.
Metal Sonic can quickly turn the tides with his different forms, like Neo Metal Sonic, but when he becomes Metal Overlord, that's going to be a big problem. However, Bowser can also become Fury Bowser to even up the playing field, and when he becomes Fury Bowser, he can launch home spikes from his shell and use his fury blast to destroy anything he comes across. But Eggman does have many personalized mechs to counter as well. His Eggmobile can fit inside the Death Egg Robot, Egg Emperor, and other mechs, but the best of the bunch is the Egg Dragoon. It comes equipped with a machine gun, ice projectile blasters, and a homing drill, making it one of Eggman's best creations to date.
However, Bowser's raw strength and toughness are a whole other story. He has foughten castles and towers and has taken their hits head-on without any issue. He has even survived cosmic-level feats as well. He had survived supernovas and black holes multiple times over in the Super Mario Galaxy series, and those things can rip anything apart, even light itself. So even Metal Sonic will have a hard time landing any meaningful damage since the Koopa King is just too tough, but strength and toughness aren't everything. Eggman's mechs have outpaced Sonic many times, and the Blue Blur has outrun a black hole for at least 30 seconds at the end of Sonic Colours, which means the Eggman Empire could be faster than the Koopa Troop.
So here are my final thoughts; Eggman might be faster, smarter, and have a better army, but Bowser's power, toughness, and stronger teamwork within the Koopa Troop will be enough to take home the win.
Results:
Bowser:
+Stronger
+Tougher
+Koopa Troop works better as a whole unit
+Fury Bowser can counter Metal Overlord
-Dumber
-Slower
-Koopa Troop is slower and weaker than the Eggman Empire
Dr. Eggman:
+Smarter
+Faster
+Mechs like the Egg Dragoon can help even out the battle against Bowser
+Metal Sonic can overpower the entire Koopa Troop by becoming Metal Overlord
+Eggman Empire is physically better than the Koopa Troop...
-... but they don't work together most of the time, always preferred to work individually
-Weaker
-Eggman never stands a chance against Bowser by himself
-Eggman Empire is no match for Bowser by themselves
Bowser FTW
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Death Battle predictions: Bowser vs Eggman
ActionMy predictions on the Death Battle between Bowser and Dr. Eggman