Rude was never going to stay once the screwjob happened, he waited and waited and then struck by being on both shows at once. You'd have to erase the whole screwjob for him to have stayed as it was the 2nd time he'd been sickened by Vince and his treatment of people.
The more interesting thread would be what if he stayed the first time...
The first time he was disgusted that he had been denied the chance to be WWE Champion in favor of Sgt. Slaughter, despite being the best heel they had. Second time was down to the screw job.
When Rude went to WCW both times he seemed revitalised, where as in DX he didn't really fit. Working with his friend Curt Hennig was a good role for him. Also at the time of his death, he was in training to return to full time action, many speculate it was this, coupled with his usual leisurely use of drugs that pushed his heart too far.
But... sake of argument he might have got the push the Bulldog did in 99...
If Rude had stayed first time:-
When Sgt. Slaughter won the title from the Warrior at The Rumble, it was already apparent that war was going to break out and Vince was going to use it, but he managed to convince Rude that he had a plan for him... Rude makes the run in on Warrior, in tandem with Savage and the two form an alliance.
This leads to the first ever "Triple Threat" match at Wrestlemania 7 with the stipulation that the lose must retire. Savage is pinned when Rude double crosses him and "steals" Queen Sherri - Savage reuinited with Elizabeth after the match. At Summerslam 91 Warrior famously tries to hold Vince up for Money before the "Match Made In Hell". While already promoting a wedding involving Savage and Liz, plan B is put into action. Warrior is "put out of commission" by The Undertaker leaving Hogan facing a 3 on 1 handicap match. Savage implores Jack Tunney to allow him to partner Hogan and is allowed to, leaving Liz unprotected. The Mega Powers are victorious over Slaughter and cronies and at Saturday Night's Main Event in September Rude claims that Liz left Randy cos she wanted to be with him, this angers Randy into a rash match which he loses when Liz kisses Rude. Rude reveals that he is in cahoots with Jake Roberts (who "brainwashed" Liz) and Jake attacks Randy with a Cobra, when Sid Justice makes a save, The Undertaker attacks him, a Tombstone to the concrete puts him "out" for a time.
Rude and Liz start to rise up the card and at Survivor Series an offshoot of the previous years "battle of survival" sees the stipulation where only one man can survive the Survivor Series. In the weeks leading up to the show, Hogan is defeated for the title by Undertaker, and then regains it. Leaving the title vacated and the "Sole Survivor" of Survivor Series would be the new champion with Hogan and Undertaker passing into the match automatically.
The teams are:-
Rude's Brood - Rude, Jake & The Natural Disasters v The Macho Madness -Savage, Big Bossman & Legion Of Doom (Rude & Jake survive)
Roddy's Rowdies - Piper, Bret, Steamboat, Bulldog v A Flair For The Gold - Ric Flair, Ted DiBiase, IRS & The Mountie (Ricky Steamboat & Bret Hart survive - Flair and Piper brawled to the back)
The Cobra Corps - Sgt Slaughter, Texas Tornado, The Rockers v Col. Mustafa, The Beverly Brothers & the Warlord (Survivors Sgt. Slaughter & Texas Tornado)
Team Nasty - Nasty Boys, Beverly Brothers & Skinner v The Bushwhackers, The New Foundation & Virgil (Survivors: The Nasty Boys)
Final Match Of Survival - For the Title -
Hogan, The Dragon, Bret Hart, Sgt. Slaughter & Texas Tornado v Rude, Jake, The Nasty Boys & The Undertaker. Rude quickly eliminate's Slaughter, Hogan takes out the Nasty's in quick succession while the Undertaker demolishes the Tornado. This leaves Rude, Jake and Taker v Hogan, Steamboat & Bret. Bret beats Jake with the Sharpshooter before being counted out after a DDT on the floor. It is now 2 on 2... Hogan is disqualified when Sid returns and rolls Taker into the casket he had planned to put Hogan in (as Sid is not in the match, it is "outside interference" and Hogan is eliminated) As Hogan and Sid argue and push each other outside the ring, Steamboat tries to capitalise on the distraction by catching Rude with a flying bodypress. Rude rolls through and with a handful of tights wins the match and the title!
Rude is bullish as Champion, boasting of how he beat 3 former champions in one night and telling fans to keep the noise down to see what a real champion looks like. In the run up to the Royal Rumble, Hogan and Sid continue their rocky friendship and battle against The Undertaker. Bret Hart is challeneged to an IC title match by Jake Roberts and Randy Savage and Rude battle over the countries house shows for the title. In late december, things are thrown for a loop... During one of Rude's intro's, Ric Flair interrupts and says how he has heard him say he is a real champion now for too long and his "real worlds championship" means HE is a real champion, that Rude had beaten someone "he had already beaten" and to be the man you have to beat the man.
Rude tells him the only way he is getting a title shot is to win the Rumble, but that there are a lot of guys who want him and Flair doesn't have the heart.
At the Rumble, Jake wins the IC title (he drops it to Piper the following night) which comes down to Flair and Sid with the same ending as the real 92 Rumble. Flair entered at #1 and has defied the odds the odds and throughout the match slowly turns face. Hogan and Sid have exploded into a feud and Rude is so angry that he makes the mistake of slapping The Undertaker for not making sure the job was done. This costs big time him with a Tombstone and a loss to Savage by DQ when Taker interferes on him, turning face.
From there, Flair beats Rude at WM8, Hogan beats Sid, Undertaker beats Jake and Bret beats Piper and 5 years on Randy Savage and Ricky Steamboat have another classic for the #1 contendership with Savage taking the win.