Bart and Lisa Simpson vs. Albert Einstein

Bart Simpson is traveling at ½ the speed of light, when he passes Lisa Simpson, who is standing still on the planet earth. Each child has a synchronous timepiece which is triggered to display 0:00:00 (h:m:s) as Bart passes. Simultaneously a laser is emitted from Lisa’s location. Bart and the laser are on a race to the sun.

At 0:04:00 on Lisa’s timepiece Bart passes ¼ AU (AU = the distance from the earth to the sun) and the laser beam passes ½ AU. When Bart passes ¼ AU he observes the readout on his timepiece. Much to his chagrin it does not read 0:04:00 because of Bart’s speed, the diminishing effects of earth’s gravity and the increasing effects of the sun’s gravity. Is there any way to calculate the read out on Bart’s timepiece? I am going to have to estimate the readout on Bart’s timepiece would read 0:02:00, because it would be consistent with the span between his position and the position of the forward most end of the laser beam.

Do LaGrange points have any significance in these calculations? Would Bart’s mass increase at ½ the speed of light? Is there any way to graphically display the topography of the warped space/time the travelers would be subjected to?
At 0:08:00 on Lisa’s timepiece Bart passes ½ AU and the laser beam passes 1 AU. When Bart passes ½ AU he observes the readout on his timepiece I am going to have to estimate the readout on Bart’s timepiece would read 0:04:00, because it would be consistent with the span between his position and the position of the forward most end of the laser beam.