Had Edward III's son Lionel, Duke of Clarence, had a son, could this have made the wars a struggle between 3 rather than 2 rival houses?
No, because the House of York didn't draw their claims from the direct male line going back to Edmund of York; they drew it through the female line, which lead back to Lionel.
The thing is, Henry Bollingbroke
was Richards heir if you went by strict salic, male-line inheritance. Roger Mortimer, as you will be able to discern from the name, was not a Plantagenet. So if you had a true, direct descent, Plantagenet, male line heir, then Bollingbroke's claim would have been substantially weaker - nay, he wouldn't have had a claim.
In all likelihood, if there had a been a usurpation of Richard here, then it would have been from Lionel's hypothetical son, who would have been Richard's actual heir in this situation - ergo, no Wars of the Roses. Or at least, not a flying fig of a real claim by anyone else to the throne.
If Bollingbroke or Gaunt or anyone else still usurped the throne with Lionel's son still around, then they would be in big, big trouble, probably with an imminent civil war on their hands.