According to This official FAQ page:
Q25. When can service animals be excluded?
A. The ADA does not require covered entities to modify policies, practices, or procedures if it would “fundamentally alter” the nature of the goods, services, programs, or activities provided to the public. Nor does it overrule legitimate safety requirements. If admitting service animals would fundamentally alter the nature of a service or program, service animals may be prohibited. In addition, if a particular service animal is out of control and the handler does not take effective action to control it, or if it is not housebroken, that animal may be excluded.
Q26. When might a service dog's presence fundamentally alter the nature of a service or program provided to the public?
A. In most settings, the presence of a service animal will not result in a fundamental alteration. However, there are some exceptions. For example, at a boarding school, service animals could be restricted from a specific area of a dormitory reserved specifically for students with allergies to dog dander. At a zoo, service animals can be restricted from areas where the animals on display are the natural prey or natural predators of dogs, where the presence of a dog would be disruptive, causing the displayed animals to behave aggressively or become agitated. They cannot be restricted from other areas of the zoo.
This seems to indicate that the zoo exception exists because it would “fundamentally alter” the nature of the zoo. I don't think that it would “fundamentally alter” the nature of a store with a mascot cat for the cat to be temporarily restricted to the upstairs living quarters or other area away from the customer areas where a service dog must be permitted to accompany a disabled customer.
In the FAQ document Service Animals under the ADA from the Great Lakes ADA center, on page 7, it is said that:
In General Covered entities must modify policies,
practices, or procedures to permit the use of a service animal by an individual with a
disability in any area open to the general public, unless the entity can demonstrate
(1) that making such modifications would fundamentally alter the nature of the
entity’s goods, services, facilities, privileges, advantages, or
accommodations, (2) the safe operation of the entity would be jeopardized, or (3)
such modifications would result in an undue financial or administrative burden.
28 C.F.R. §§ 35.130(b)(7), 35.136, 35.150 (a)(3), 35.164, 36.301(b), 36.302 (c)(1),
DOJ commentary suggests that Congress intended the ADA
to allow service animals the “broadest feasible access” to public
accommodations and public entities and to avoid unnecessarily separating service
animals from from their owners. 28 C.F.R. pt. 36, App. C.
Covered entities that have blanket policies or practices that exclude service animals
may be subjected to court orders or settlement agreements requiring modification of the relevant policy or practice.
On pages 8-9 of the same document it is said that:
It is the entity’s burden to allege and prove the existence of a fundamental alteration. The outcome of such defense will depend on the distinct
facts of each case.
The document cites Lentini v. California Center for the Arts, Escondido, 370 F.3d 837 (9th Cir.2004), in which (according to the document) an attempt to exclude a service dog which had previously barked during musical concerts was not allowed, and a "fundamental alteration" defense was denied at both the distinct and circuit court levels.
The Great Lakes document also cited Johnson v. Gambrinus Company/Spoetzel Brewery, 116 F.3d 1052 (5th Cir. 1997), in which a brewery attempted to exclude a service dog for a tour open to the public, Citing FDA regulations in support of a "no animals" policy, and argued that to admit animals would constitute a "fundamental change". The Fifth Circuit ruled that the brewery must modify its policy, and awarded damages under a parallel state law.
This seems to confirm that the "zoo exception" is limited to a situation where the animals on display are a "fundamental" part of the operation of the facility, as is surely the case with a zoo, but would not be the case in a store as described in the question.
I have not found any case law directly on point. If the issue has been in court, it probably has not been taken to the appeals court level where opinions are published.