Updating project to allow Swift and Objective C++ Interop

Recently, in Swift 5.9 I think support was released for interoperability between Swift and Objective-C++/C++. I'm currently working on a project that has a lot of Swift, Objective-C and Objective-C++/C++ and not being able to interoperate between Swift and C++ has always been a sticking point requiring an array of messy wrappers. Because of this I updated my project's Swift version to Swift 6 with hopes to avoid this and fully interoperate.

The last setting I needed to change (I thought) was C++ and Objective-C Interoperability from "C/Objective-C" to "C++/Objective-C++". Though, doing this though created numerous errors of the variety of

/Users/---/Bridging-Header.h:7:9 failed to emit precompiled header '/Users/---/Library/Developer/Xcode/DerivedData/Project-akhczzqbkaltwdgzmryrlhcpnduq/Build/Intermediates.noindex/PrecompiledHeaders/Bridging-Header-swift_3CPL2KPLMYXOU-clang_35SMM50I2WP8K.pch' for bridging header '---/Bridging-Header.h'

Note: sensitive file paths have been omitted and replaced with ---

For some reason, my bridging header just wouldn't emit the headers and I wasn't really sure what was wrong. I've tried playing with various related build settings to get rid of these errors but all to no avail. Is what I want actually possible and if so, how do I make these errors stop?

How do I properly update my project to avoid these errors and fully support Objective-C++/C++ interoperability?