How do I mask lines so that I can still edit them and retain end caps for the masked drawing?

[sssnole]

I am trying to mask this drawing, but the lines where the mask is are cut off abruptly. Is there a way to edit a shape created with lines so that the lines still retain properties like end caps?

[Myra Ferguson]

Here's some more info on the Shaper tool in case that might help Build new shapes with Shaper and Shape Builder tools in Illustrator