Does being regressive make a policy bad? While I agree, it is a point against, it's not the final decision point. A policy should be looked at in terms of its larger context.
While it's hard to justify putting additional burdens on those with less, a policy can be enacted within a context of policies that overall lighten the burden. It's beneficial to do so when the policy has net benefits. For example, a basic carbon tax is regressive; the poor as a percentage of their income are more dependent on gasoline and energy. But if a carbon tax is enacted within the context of a equal direct cash payment, the overall effect is progressive.
It's also important to recognize that distribution can be the smaller question, even if it's always a big question. Climate change is a problem that will affect all. It affects many today, and the effects will only continue to grow. Those effects aren't on the same timescale though. The costs of a carbon tax will start immediately, the benefits will take time to emerge.
It's this non-equivalent comparison that is the most complex part of trades. It might be short term for future term. Or it might be money for benefits not expressed in currency.
Using a sugar tax as an example, if the purpose of the tax is to incent better behavior, to "nudge", then the benefit of behavior change, even to lower-income groups, can exceed its direct cost. This is what Hunt Allcott of New York University, Benjamin Lockwood of the University of Pennsylvania and Dmitry Taubinsky of the University of California, Berkeley looked at in Regressive Sin Taxes, with an Application to the Optimal Soda Tax. The net health benefit of the altered behavior is more valuable to individuals than the money lost. While it would be regressive from a money perspective, it would then be progressive from a well-being perspective.
But are policies already tilted toward putting too much of a burden on the poor, or even middle-income Americans? I think so, which is why I'd always support pairing changes that are regressive in money terms with changes that are progressive. But in both cases I'd search for changes that have positive social benefit overall.
The overall social benefit is another aspect important to acknowledge. While it's tempting to try and design a policy that has no negative impact on the bank accounts of those with the least, it's not always possible to achieve the objectives in that way. Wealthy individuals are not the only ones who can create harm. Due to numbers and lack of choices, you'll often find that a significant amount of harm is done by less wealthy individuals. Incenting changes in behavior is valuable. Abandoning efforts to change actions because the policy that fits is regressive blocks important action.
Compensating within the policy itself sounds attractive, but it's often difficult, impossible or self-defeating. For example, creating an exception for low-income individuals to a pay-by-the-pound refuse program wouldn't incent change where it's needed most.
Instead of making an exception to this policy, do something that overall assists the less wealthy, such as housing vouchers. It might initially seem badly designed to pair housing policy and refuse collection. But the point is to have the two policies separated by a wide enough gap that they don't affect each other.