Lee tried to enter the fray three times way before Appomattox and his men wouldn't let him. The idea of a suicide ride crossed his mind - he mused to his aides that he could go out there, just him, Traveller and the Confederate flag, and jog along the enemy lines until they closed his business. Again - his men would get killed preventing him from doing it. Any time Lee wanted to get into the action himself, they stopped him!
One thing Grant was worried about was a guerilla war, which would have been the only way to keep from losing - never winning but never losing. This was suggested to Lee and had he given the word, his army would have vaporized into the hills right before the Union forces. Grant would have spent the rest of his life fighting them then. Sherman believed Forrest would do this, which is why he made it clear without saying so that Forrest had to be killed. The cavalrymen were his biggest fear. It was not desirable, however, for Lee to bite the dust in combat at this point. That would have made many of his men turn renegade and vow to fight forever.
Having an Alamo last stand fight at Amelia Courthouse might have made some sense but Lee could not see the gain in getting people killed who had been the die-hard loyal veterans of many battles. This would have been a waste of dedicated lives that the South would need to rebuild - these soldiers couldn't be murdered, and that is how Lee would have seen such a fight. In the west, there were still two barely viable armies in the field - Taylor's and Kirby-Smith's as well as remnants of Hood's army. The cavalry was in better shape in the Trans-Mississippi with JO Shelby still a viable force. Taylor was finished but there was a chance Forrest might continue to fight - he told them anybody who thought the war should continue was a madman who needed to be sent to an asylum immediately. Lee would have agreed with him! George Thomas told Forrest Mississippi and Tennessee would be a hundred years buried if he decided to keep fighting - but he didn't need to make the threat. Lee knew it was over when it was over and...it was over. Further fighting meant further destruction of what he had been fighting for.