Eggman has some other decent things like Power Absorption, BFR, Type 6 Immortality, Time Stop, Planetary Roboticization, and even has some nuclear warheads. You should read his profile (specifically the "Powers and Abilities and "Notable Attacks/Techniques" sections) to check out everything he has at his disposal:
Bowser would have no way of coming back from BFR, which is as simple as Eggman shooting him with a beam from a glove. Even if Bowser destroys the Death Egg, Type 6 Immortality will only have it as an inconvenience. Mind controlling waves from the Psionic Emitters also say hello, to which Bowser has no resistance to. The Time Immobilizer will also be a detriment towards Bowser.
Eggman takes this via sheer versatility and Type 6 Immortality if things go south.
I know nothing about this Eggman, so there's likely several counters, but what's stopping Bowser from teleporting to him and just tapping? Bowser's the vastly superior fighter in every sense of the word.
Hm? Not fighting hand-to-hand/without inventions doesn't indicate a lack of combat skills.
"Calculating and deceptive, Robotnik was able to successfully win the Great War, overthrow the Kingdom of Acorn, and rule for over a decade. During his reign, Robotnik masterminded numerous schemes to undermine the Freedom Fighters." - Eggman's intellect section
Plus Robotnik has fought Sonic in close corners combat, who is listed as "Combat/Tactical Genius"
Bowser definitely does start with Transmutation several times in-character, or something similar/sufficient. At the beginning of SMB, he transmuted all of the Toads to bricks, invisible blocks, or horsetail plants. He's done this on some other occasions, like turning a bunch of random people in Puzzles and Dragons to bricks, like Paratroopas into the red bricks. He transmutes Koopas to toads in Mario Party 2. He has started with Sealing before, too, such as when he stole the Star Rod and rewrote the whole story and then sealed the Star Spirits away, or when he used the Power Stars to seal away the Toads.
Is there anything to indicate Bowser built all that stuff himself?
Also, you should source Mario/Bowser fighting in millions of battles, personally having 100s of years of combat experience, and fighting peeps with billions of years of combat experience.
I meant to post this one for the million battles things. And yeah, I'm the same scene where Bowser was said to have an IQ of 9800, he created life by mixing a bunch of chemicals together. And there are implications that Bowser made those things in the links given. And even if you say Dimentio doesn't have billions of years of combat experience despite being that old, Antasma and Zeekeeper fought against each other in war for millions of years, and Mario fought him.
"After Mario trounced Bowser for the millionth time" sounds like a figure of speech. Similar to the saying "I've told you a million times" in order to convey repetition.
Artificially producing life isn't wildly more impressive Eggman's accomplishments.
I didn't see any implications that Bowser created that stuff. Just because it's referred to as "Bowser's so and so" doesn't mean that he built that stuff from scratch. All that conveys is that he owns that stuff.
I've like a link to confirm that Antasma and the Zeekeeper warred for millions for years straight.