Unfortunately they can retroactively charge you for the premium that was due. You were notified of the premium increase in a timely manner and they can correct the billing that was mistaken. You may be able to arrange paying the additional money in separate monthly amounts, but that is something you would have to discuss with the provider.
Yes, they are allowed to retroactively charge you a higher premium. The better question you might want to ask your self is do you really want to risk cancellation of the policy for failure to pay the premium amount? They did inform you that the amount was going to go up to $533, but they failed to bill you at that amount. Then they figured it out later on. I think you should just pay the amount of the premium.
This is not to be considered legal advice and we have no attorney/client relationship.
For the first 18 months of COBRA you can be charged 102% of the actual cost of the insurance. Make sure you are not paying more than that. If you were not billed enough then you can be charged the corrected amount subject to that 102% limit. You can lose COBRA for not paying the correct amount so work this out quickly.
More facts need to be known so do not rely on comments here until consuting with an attorney that practices regularly in this area of law.