This is a difficult and emotional predicament. My opinion won't make me popular, but here it is... bear in mind that I love animals, especially cats and dogs. The aspirin regime seems benign and affordable enough. From what I gather of your finances (the fact that you would work overtime to pay for the $400 test), I would not do the test, nor look into what might be potentially an expensive drug regime. Those $400 are best saved for a rainy day. Your cat's life might be shorter, but you also need to consider unexpected changes in your situation (health, employment) that you cannot predict and that require financial slack.

@badger: The comparison to a clutch is not a fair one. You mentioned you might changed the clutch because you had to... No car, perhaps translates into no means of transportation to work and replacing the clutch is a cheaper option to replacing the car?