Hi schannell,
yes this would be pretty easy with Reaper. Before you record you should clap once or twice to later have a reference point for syncing the videos.
Once recorded, only thing to do would be loading the videos into Reaper at different tracks, aligning them exactly using the clap as an audio peak and/or in the video as visual marking. Don't forget that the video track at the top of the stack always covers the others below. So make sure to mute a video if you want to see whats going on at the video track below.
Then import the recorded audio on a separate track to also align it to the video via the claps. If done, you can mute the audio of the videos by decreasing the item volume to -inf or by using the video properties (F2/video properties)to turn it completely off. (No idea if this even spares some cpu.)
Now you can add a video processor to every video track's FX chain and choose the easiest way to blend over by using the preset:
Item fades affect video
It lets you use the item fades as opacity fade. Cut the top video at certain areas and use the fades to blend to the video below in the stack. Cut out more areas and fade the hell out of your videos.
Don't forget that at least one track with video needs to be there else you get a green screen (or in the next Reaper updates a black screen).
Hope that will help you to get started. If you have questions for more advanced tasks join the video forum and start a discussion.
Greetings
Eli