Is a double quasi-binary system valid? I was messing around with 4-body problems, and discovered a system where a close binary and a regular binary pair orbit each other. The regular binary gets disrupted every few orbits and one star orbits the close binary, while the other orbits farther out. Then, as the outer star reaches periastron, the inner star gets captured by the outer star and the cycle begins again, shown in this video (the stars here are color-coded).
Is this possible and is stable in the long-term (without any external perturbations), and have we discovered anything similar to this?
As far as I know, all known old multiple star systems with stable orbits have highly hierarchical structures.
Binary star pairs in those systems orbit much closer to each other than they do to other binary pairs or single stars within the system.  In such a system it would be very unlikely for a star which is part of one binary to be temporarily captured by another binary and then return to its original binary.
Exceptions are multiple systems of very young stars which have formed recently.  The orbits in those systems are thus very new and have not yet been regularized.  It is believed that the stars in those systems will eventually move into hierarchical orbits or be ejected from those systems.
So I think that your system seems very improbable.  However, you won't know for certain unless you can run highly accurate simulations covering billions of years of evolution of the orbits in your system.