Not sure. The behavior you posit sounds like a virus, but the regular cells would lack the machinery to translate the artificial genes. Defenses against bacteria are mostly directed against surface proteins and these proposed organisms would have regular proteins

Or a prion - not really a virus, but an agent that causes the body to misbehave, almost by accident.

A misfolded protein? That sounds more likely with this type of DNA, doesn't it?

Considering how rare prions are I don't consider it likely that an escaped artificial DNA organism would accidentally produce one

Organisms with the new base pairs probably wouldn't have so much of an advantage that they completely replace all life that currently exists.

However, it does raise an interesting question if some future species would be able to figure out evolution and the origins of life, since there would literally be intelligently-designed organisms running around.

Pathogens are not fought by attacking their DNA. It's not going to make any difference, except that it's harder to breed it with anything else.

Isn't this exactly how bacteria fights viruses? That is - by cutting known dna patterns using restriction enzymes. https://en.m.wikipedia.org/wiki/Restriction_enzyme

And since p and z are synthesized in the lab how would this escaped organism eat?

It would have to have evolved or be designed to be able to synthesize them.

https://en.wikipedia.org/wiki/Nucleotide#Synthesis

Life finds a way.

Why hasn't life found a better way to fix nitrogen yet? Nitrogenase is a godawful enzyme... Wastes three hydrogen gas molecules for each turn of the crank.