I understand that I will have to either have to
a) include binaries & sources of the used libraries with my software package
This is not correct. You can redistribute binaries without the sources and the sources without the binaries. In all versions of the LGPL, you need to eventually redistribute the corresponding source code if you only redistribute the binaries. The GPL 3.0 section 6 explains how you can handle the redistribution of the corresponding source code when you redistribute a binary (and this is in the GPL and not the LGPL 3.0 text, the former being incorporated by reference in the LGPL 3.0 text) and there are at least six different ways.
b) ship my software without the libraries and tell the user to get them themselves
This is not correct. You can include the libraries alright and redistribute the corresponding source code for the LGPL-licensed library either bundled or separately from your software product. Depending whether or not the LGPL-licensed library has been modified and how your code interacts with this library you may have additional obligations and the copyleft may extend in some cases to your own code calling this library.
I am wondering/unsure if my including a utility that downloads the required binaries onto the user's computer, instead of having them do it themselves, would be permissible/compliant to the license terms of these libraries?
This may be OK in some cases but will never comply with the spirit of the license (beside being a major pain for your users)... Furthermore, if you are for instance using a programming language that compiles to native code such as C/C++ and are dynamically linking with the LGPL-licensed library, you are de facto incorporating header includes in your own binary which makes the separate user installation of the library a rather moot point.