There are some bugs contaning or exuding toxins which could in theory harm mammals if ingested in sufficient lots. Examples are ladybirds or green shield bugs.
But this usually won't happen. Toxins produced by bugs have a evolutionary purpose. It is to prevent predators from eating the bugs. So what is the best strategy for a bug to prevent being eaten by using a toxin?
The toxin could harm the predator silently by e.g. damaging organs like liver or kidney. If the predator had eaten enough bugs it would fall sick and die eventually
The toxin could kill the predator instantly.
The toxin could taste extremely bad or have an instant irritant effect. Only a very prolonged consumption would lead to substantial health risks of the predator.
The best strategy is Nr. 3 for different reasons. Nr. 1 is the worst. Nr. 1 neither prevents the bug from being eaten nor trigger a learning process at the predator. Probably a big bunch of bugs will be eaten by a single predator and liftetime of the predator may be or may be not reduced substantially. Especially if the bug is a rare species.
Nr. 2 is slightly better but will be easily rendered useless as most predators tend to breed in case of sufficient food. A evolutionary effect (predators starting to mitigate the bug) will only take place, if the bugs can be discriminated by the predator from other species. This is usually achieved by warning coloration.
Nr. 3 is the best, because it usually takes only one contact of a bug with the predator to prevent any further attacks from that individual. It even opens the possibility of survival by the individual bug because it is released before being swallowed. The predator will survive and occupy the ecological niche, preventing other predators with similar habits from preying on the bugs. If the social skills of the predator are high enough it will even pass the meme to its offsprings. Accompanied by a warning coloration like on ladybirds, this option maximises the likelyhood of beeing not eaten by this kind of predators.
This is why most allomones produced by bugs primarily have an irritant or deterrent effect which supersedes the health danger of the predator by magnitude. Both, the prey and the predator profit from this strategy.
So you can safely assume your cat will taste only uncritical portions of unhealthy bugs.