First let’s describe some of the guerrilla warfare that took place before and during the civil war.

Missouri and Kansas represented some of the most brutal and violent sites of guerrilla tactics. On the Confederate side you had William Quantrill, “Bloody Bill” Anderson, and George Todd. Conducting guerrilla warfare for the Union were “Jayhawkers” like James Lane, Charles Jennison, and James Montgomery. All these men were likely heavily influenced by the violence of “Bleeding Kansas,” which occurred in their home state the decade preceding the war. Pro and Anti-slavery civilians had waged a long campaign of raiding against each other. Irregular warfare was not new to them when the war came, and this showed during their exploits at Lawrence, Kansas.

Lawrence had already suffered a bout of guerrilla style warfare prior to the war officially beginning. In 1855-56, pro-slavery militias laid siege to the predominantly anti-slavery “Free-Soil” town. Although only one person died, it set the stage for what would happen just under a decade later. During the raiding and civilian fighting that continued through the early years of the war, the Union commander in the area, Thomas Ewing (Brother-in-law of William T. Sherman), rounded up the wives of some of the guerrillas (Including the previously mentioned William Quantrill) who had been providing lodging and aid to the fighters. Ewing placed them all in a “women’s prison.” Unfortunately for them, the building they were being held in subsequently collapsed, and five of them perished.

This incensed the confederate raiders. They were already infuriated by the Union raid on Osceola, Missouri where Union partisan James Lane had raided the town, and held “Drumhead court-martials” for 9 men (kangaroo courts essentially), and subsequently executed them. The Confederate partisans had assembled a band of 450 men led by Quantrill. The subsequently killed 182 men and boys, and razed 185 buildings in Lawrence. Union cavalry arrived eventually, but too late to stop the carnage or capture any of the raiders. Ewing responded with General Order No.11, which banished many civilians from different Missouri counties on the border of Kansas. It had little effect, and the raids continued

Now is all this really what you might think of as guerrilla warfare? I would say yes, particularly as it relates to the Union side. Federal troops were already spread thin in Kansas, and they were unable to keep up with the hit and run tactics used by either side. Although these instances of guerrilla warfare were not necessarily aimed directly at the Union or Confederate armies , they still had the expected results of bogging down said militaries.

Now on to your questions:

How serious were these plans

Not very. The suggestion to conduct a guerrilla warfare came either just before Lee’s surrender (according to McPherson) or just after (according to Hsieh). Either way, both historians describe his response as being promptly in the negative. Here is his reply to the officer suggesting the guerrilla campaign, Brigadier General Edward Porter Alexander:

"...the men would have no rations & they would be under no discipline. They are already demoralized by four years of war. They would have to plunder & rob to procure subsistence. The country would be full of lawless bands in every part, & a state of society would ensue from which it would take the country years to recover...And as for myself, while you young men might afford to go to bushwhacking, the only proper & dignified course for me would be to surrender myself & take the consequences of my actions."

As you can see, Lee had little palate for such a tactic at that point. He believed such a campaign would just lead to more dead Confederates. Not only that, but he dreaded seeing any more destruction fall upon Virginia. Some historians have debated whether his love for Virginia affected his campaign decisions throughout the war, but McPherson points out that overall strategy was the responsibility of President Jefferson Davis.
Nevertheless, the point of contingency in the war had long since past. There were a variety of major points of contingency in the war: Antietam and Perryville in 1872, Gettysburg, Vicksburg, and Chattanooga in 1863. However, the Confederate losses in 1864 in the Shenandoah and the capture of Atlanta were the true final points of contingency. Even if Lee had wished to fight a guerrilla war, it is unlikely it would have faced much success given the state of the Confederacy in 1865.

And how significant was Lee's role in this moment?

Considering Lee clearly declined engaging in a partisan war: pretty significant. But I suppose more importantly we might ask why Lee made this decision. Obviously I have already previously pointed out that operationally and logistically it was unlikely to prove to be a successful campaign. According to Wayne Wei-Hsiang Hsieh, Lee's institutional grounding in the antebellum Army remained influential throughout his career as the top Confederate general. His education at West Point was particularly important. Lee had grown accustomed to the regulations and discipline of a uniformed army, and thought the use of guerrillas to be unbecoming of the Confederates. If he could not fight in the style of war that he had been taught, then he would not fight at all.

Sources:

Hsieh, Wayne Wei-Siang. West Pointers and the Civil War: The Old Army in War and Peace. 2009

McPherson, James. Battle Cry of Freedom: The Civil War Era. 1987.