By happy accident, I stumbled across this story while looking for something else. It's "Marooned on Planet Earth" (1985) by Thomas Wylde. The only publication listed in ISFDB is the March 1985 issue of Isaac Asimov's Science Fiction Magazine, which must be where I had read it originally.
My memory was fairly distorted, so I'm not surprised that nobody recognized it. The fact that the protagonist is a normal person seeking the transformation voluntarily is established in the first lines:
"Why do you want to be an alien?"
Jeff Schuster smiled at the question. Wasn't it obvious?
The doctor leaned forward, perhaps to get a better look at Jeff's
smile.
Jeff Schuster cleared his throat, and the doctor's head made a small
anticipatory nod. But Jeff remained silent.
After a moment, the doctor said, "I understand. You feel helpless, out
of control, yes? Adrift in a lifeboat, perhaps. Rudderless. Becalmed
under a broiling sun, yes? You feel your life slipping away, dribbling
out, escaping into thin air. You want a purpose for living. You want
life to be an adventure. Yes? Is that it?"
Jeff's head had started to nod almost from the first and was by now
rocking up and down emphatically. "Yes!"
The treatment involves being given the illusion that the patient is an alien on a mission:
..."Sign both copies. In four months you'll return for an evaluation.
Don't worry about remembering, it's all built in." He quickly unfolded
Jeff's hundreds and gave the bills a snap, peering at them briefly in
the window's light. "Very good." He glanced at the signed forms,
pulled a carbon for Jeff, and shoved the rest into a drawer. "Relax!"
he said. "The next thing you know, you'll be waking up in bed tomorrow
morning. And when you do--"
"I'll be an alien."
"Guaranteed."
"With a secret mission."
"Absolutely."
The process is some sort of hypnosis:
The alienist's arm shot into the air and wiggled like a drunken snake.
Then his fist opened and Schuster was staring into a battery-powered
Hypno-lite®. Brilliance swelled to a fascinating conclusion.
And that was the end of Jeff Schuster.
Shortly after waking up, the "alien" Xgglm sets about improving the situation of his host:
Next he examined his unprepossessing host. Another mess.
Later, when he had consulted information storage devices in the nearby emporium called a "library," Xgglm confirmed that his adopted body was sheathed in excess poundage of subcutaneous gelatinous fatty cells. Fortunately, there was plenty of easily understood advice concerning the removal of this unhealthy mass, and Xgglm set about the task with enthusiasm.
He also learned that the brain of the beast--which he was forced to use--contained a vast and easily manipulated data storage and retrieval system, virtually empty at present. He commenced filling it with the contents of the library, for it was impossible to know this early in the game what information might lead to his escape.
Xgglm concluded he might require an added measure of strength, so he put himself on a program of exercise and muscle building. As the body neared the maximum of its potential, Xgglm took it outside to cooks its skin in the ultra-violet--a "tan" being highly prized by the natives.
His mission is apparently nothing more than the need to get off the planet, which will require a ship that can't be manufactured with current technology. To that end, he seeks out a local UFO enthusiast club, where he meets someone that he knew from high school:
The group called itself the Extraterrestrial Propulsion Society, and
was concerned with the location of alien spacecraft. There was no
doubt--judging from the incredible accounts--that these flying saucers
(as they were frequently called) were precisely what Xgglm sought.
Within minutes he was on the Hollywood Freeway in his new Corvette. He
got off at Roscoe, and was soon traveling east into the hill of
Tujunga.
It was noon and the sun was hot, the air spicy with sage from
the chaparral. He pulled up in front of the house listed in the
newspaper.
...
"Who?" said Xgglm, rapidly searching his memory files. Jeff Schuster... Jeff Schuster...
"Beverly Newton," she said. "We were in high school together. But then you were... God, you look great now."
Xgglm grinned the grin he learned from Indiana Jones.
The group has a rumor of an actual downed spaceship that supposedly crashed into a mountain but left no trace. (This is already established to have occurred.) The protagonist is unafraid when the craft is discovered and enters it hoping to hijack it for his own purposes. He manages to do so, bluffing the UFO's pilot (called Bob in the story) that a tire gauge is some kind of weapon.
Bob self-destructs after ordering the ship to leave Earth, and the light of the true alien's immolation triggers Jeff's return to his normal self. Far from being transformed for the better, he immediately reverts to old habits of thought:
Then he remembered the alienist and the hypno-therapy and the
startling morning he woke convinced he was Xgglm and marooned on the
planet Earth. Well, he had made it off the Earth. Congratulations.
...
He closed his eyes and tried to force himself back into the persona of
the alien whose name whose name he couldn't pronounce, to recapture
the cosmic sense that the world was new and mysterious and full of
promise, that life was an adventure-- And when he opened his eyes he
was... Jeff Schuster.
Naturally he fainted.
The story is only about 12 pages long. It is available at the Internet Archive, for those interested.