Thanks everyone!
The matches that I am looking for (which JohnGG correctly highlighted) are:
ATGGTTTCTCCCATCTCTCCATCGGCATAA
ATGGTTTCTCCCATCTCTCCATCGGCATAAAAATACAGAATGA
ATGGTTTCTCCCATCTCTCCATCGGCATAAAAATACAGAATGATCTAA
ATGATCTAA
Comment on Re^2: Using Recursion to Find DNA Sequences
To get the shortest-to-longest length ordering from the approach I outlined here, just use a ? "lazy" modifier on the quantifier of the \w* "padding" sub-pattern making it \w*?: qr{ ATG \w*? (?: TAG | TAA | TGA) }xms;