I was assuming that "feeling bad when bad things happen to someone" is part of the definition of what it means to care about someone. And I'm naturally reluctant to choose to not care.

You still have preferences in addition to emotions. Say you have a strong preference for bad things not to happen to someone. Then you do whatever you can to prevent bad things from happening to them, and if something bad does happen to them, you help them out to the best of your ability. In my book, that counts as caring about someone. Not caring would mean that you didn't do anything to stop them from experiencing bad stuff, nor did you help them out if something bad did happen to them.

Now people have various definitions of caring, and some probably do think that "feeling bad if something bad happens to someone" is required for genuine caring. But I would disagree. From an evolutionary point of view, emotions exist to motivate behavior. If you behave like a caring person would but don't feel bad, then in reality you care more than someone who feels bad but doesn't actually do anything. And if you end up feeling bad, then that may distract you and cause you to make worse decisions or temporarily paralyze you, which reduces your ability to actually help. (Also, evolutionarily, feeling bad about someone suffering probably also acts as a costly signal: if they're hurt, you suffer, so they have unfakeable evidence of you actually caring and not just pretending to care. But you have no need to prove to yourself that you care in such a perverse way, and you can also prove your caring to them with your actions.)

From a consequentialist point of view also, what matters is your actual behavior. The less time you spend feeling bad, the more time you can spend things that actually make people better off.

Also, not feeling bad doesn't mean that you can't express sympathy. You can still honestly say things like "I wish things got better for you". For most people, it's the notion that you don't care what happens to them that is bothersome. You can show with both your words and actions that you do care, that is, have a preference that things go well for them and are prepared to spend time and effort to help them out if necessary.

An important note when it comes to sadness: to some extent, it seems like sadness is the appropriate response when e.g. someone close to you dies. "You're never obligated to feel bad" means that you have no moral obligation to suffer, it doesn't mean that you should try to push away or suppress negative emotions. Remember, trying to do that is exactly what causes the negative emotion-related suffering in the first place. So if you feel sad about someone dying, say, that's perfectly appropriate! It's what your brain needs to do in order to adapt to the loss. But even then there's no obligation to suffer from the sadness.

oops, I just realized... if the rule is "only have emotions about situations that were within my immediate control", and you know that the other person will feel upset if they don't see you feeling bad about their situation, then that counts as something that's within your immediate control... though something about this seems like it doesn't quite fit... it feels like I'm interpreting the rule to mean something other than what was intended...

In principle, that's correct, though you're right that it's a bit different from what I was intending. Here's something closer to what I was thinking about.

Here, I'm going to use the expression "feel bad" to refer to feeling emotions that are usually considered negative. I don't mean that one should actually find them aversive or suffer from them. More on this below.

Suppose something bad happens to person X, who you care about. The bad thing wasn't anything you had control over, so you have no reason to feel bad about it. But now you have a chance to help X. Whether you help them or not is something you do have control over, so if you do help them, you should feel good about it.

But suppose that you fail to help them. Now it may or may not be appropriate to feel bad, depending on why you fail to help them. For instance, maybe you are driving to their home, but on the way there your car breaks down. Presuming you hadn't ignored clear signs of an immediate breakdown or otherwise clearly neglected the maintenance of your car, then it breaking down wasn't really under your control. This prevents you from helping them, but it still isn't something that you should feel bad about. Feeling bad is a feedback mechanism to teach you lessons about what you did wrong, and there are no useful lessons to be learned here.

You should only feel bad if you failed because of something that was under your control. Maybe you were going to take a bus to them, but got stuck online and missed the last bus. Or maybe you drove your car carelessly and got in an accident. In that case it's okay to feel bad, as your behavior mechanisms need feedback.

Still, even if you feel bad, ideally you shouldn't suffer from it. Blaming yourself accomplishes nothing. Your attitude should be "okay, I now made a mistake, so I'll gladly embrace this momentary pain and be happy over the fact that it will teach me to act better in the future". This is always a good mindset to have, because it will increase the odds of you actually acting better in the future. Being prepared to accept any pain without needless guilt is good, as it makes it easier to internalize the actual lessons of your mistake without wasting energy on needless suffering. And if the attention-allocation theory of suffering is true, then suffering is always needless, because it means that your brain is wasting energy and resources being pulled in opposite directions.

If you screw up and feel bad, you may think something like: it's a bad thing that I screwed up, but it's also a good thing that this pain is teaching me not to do it anymore. Now I'm going to feel good and happy about this enjoyable pain, because it means I'll do better in the future.

But be careful not to mix in feelings of martyrdom, self-pity or anything like that. The lesson is not "I'm a terrible person and I deserve all this suffering I got so I'm going to revel in it". Nobody ever deserves to suffer. The lesson is "I did a mistake but that doesn't affect my worth as a person. Next time I'll do better". If you're a utilitarian seeking to increase well-being or decrease suffering, that includes your own well-being and your own suffering.

Something also feels Wrong about enjoying sadness. If you happen to enjoy sadness, then you need to be really careful not to deliberately cause harmful things to happen to yourself or others, just for the sake of experiencing the sadness.

There is probably some risk of this, yes. But ideally, your behavior should be driven by your preferences. This becomes a lot easier once emotions stop being your enemy and you don't need to avoid feeling any particular emotion. When all your emotions are your welcome allies, then it's also easier to let your preferences guide your behavior in everything. That means that you've accepted feelings such as sadness as appropriate error messages that pop up when things haven't gone as they should. Then you won't be actively trying to cause those emotions, instead concentrating on seeking pleasure from doing things right.