A number of studies have been done that investigated resistance to Fenoxaprop-P-ethyl and related compounds. Most of these studies focused on resistance in crops (e.g., maize) or black grass. One of these studies (Kay et al., 1999) was particularly interesting as it focused on several possible mechanisms of herbicide resistance in Alopecurus myosuroides (Black-Grass). I quote from the abstract:
[...] Resistance at the whole-plant level correlated well with reduced acetyl Co-enzyme A carboxylase sensitivity (target site resistance) in two of the populations (one from the United Kingdom and one from Germany) but enhanced metabolism appeared to be the primary mechanism of resistance in the majority of the other populations. [...] We suggest that resistance in Lincs E1 may be explained by multiple resistance mechanisms—the expression of both insensitive acetyl Co-enzyme A carboxylase and an increased rate of detoxification. However, resistance in the population Clay from the United Kingdom could be explained neither by target site insensitivity nor by an enhanced rate of metabolism. These results provide conclusive evidence that a single resistance mechanism alone cannot explain insensitivity to fenoxaprop-P-ethyl in European resistant black-grass populations and that additional, as yet uncharacterized, mechanisms must also be present.
So based on these data, I think several mechanisms may play a role in the cool season lawn grass resistance (a species name may help to specify my, admittedly, broad answer).
Interestingly, Fenoxaprop-P-ethyl is an acetyl CoA carboxylase (ACC) inhibitor (Kay et al., 1999). Note that Haloxyfop (as mentioned in your question "How does Haloxyfop control young grassy weeds in fields of broadleaved crops?") is also an ACC inhibitor. From the answer on that question it appears that ACC inhibitors inhibit cell membrane formation and that monocots (grasses) are most susceptible and dicots less. Dicots are less sensitive because they have two ACC types at their disposal, one being insensitive to ACC inhibitors. Not surprisingly then, Bermuda grass, being a monocot is killed by this herbicide. Interestingly however, other monocots seem to survive, such as the cool-season lawns you mention.
I was not able to find conclusive evidence as to why some monocots survive ACC inhibitors. An educated guess is that resistance can be conferred by mutations in the ACC enzyme (Letouzé & Gasquez, 2001) that make the enzyme less sensitive to the herbicide. These mutations can be caused by repeated exposure to these herbicides, whereby strains become resistant to the material, much like the way in which (multi-) resistant bacterial strains emerge after repeated exposure to antibiotics.
Kay et al. Pesticide Biochem Physiol 1999; 65:169–80
Letouzé & Gasquez, Theoretical Applied Genetics 2001; 103:288-96