Is there really something wrong with using the caption package for \ContinuedFloat?

Perhaps not an answer, but it was too long to make it a comment. The documentation of the caption package says:

If you don’t find the document class you use in this section, you usually don’t have to worry: Many document classes (e.g. the octavo class) are derived from one of the standard document classes article, report, or book, and behave the same regarding captions. The caption package automatically does a compatibility check against the document class used and will give you the clear warning

Package caption Warning: Unsupported document class (or package)
detected, (caption) usage of the caption package is not recommended.
See the caption package documentation for explanation.

if such an incompatibility was detected. If you don’t get such warning everything is fine, but if you get it the usage of the caption package is not recommended and especially not supported.

If you get such a compatibility warning but decide to use the caption package anyway, you should watch carefully what side-effects occur, usually the look and feel of your captions will change by just including the caption package without options, meaning they do not look like as intended by the author of the document class. If this is fine for you, you should first specify the option style=base via \usepackage[style=base]{caption} or \captionsetup{style=base} to set the caption package into a well-defined state. Afterwards you can start setting your own options additionally and keep your fingers crossed.

Also, according to the documentation, the supported document classes are:

• Standard LaTeX: book, report, article
• AMS: amsart, amsproc, amsbook
• beamer
• KOMA-Script: scrreprt, scrartcl, scrbook
• NTG: artikel, rapport, boek
• SMF: smafart, smfbook
• thesis

So, there's no problem using caption with the standard report document class. Since you are using a derived document class, you should pay attention to the last paragraph of the quote at the beginning of this message.

One of your packages tries to load captions twice. I have the same issue because I'm loading the \subfig package, this blog post give an elegant solution:

Find the offending package and load it without the caption package.

\usepackage[caption=false]{subfig}

This has worked for me. It would really help though if you gave us your header and we could try and spot what was trying to load caption, and reduce the number of packages doing that to one.

Gonzalo's answer has explained why you can or can not include the caption package.

If you are sure you want to use it, and want to get rid of the warning, use

\usepackage{silence}
\WarningFilter{caption}{Unsupported document class}