I enjoyed Carmageddon years ago, when it first came out. Now, I'd like to install and play it on my laptop, running Fedora 35. I know that it needs Dosbox, and probably Wine, and those were both installed when I first installed Fedora. Is this going to cause any kind of conflict, or will the installer find them both and not try to install its own version?

Carmageddon Max Pack installer is for Windows only and it is prepackaged with DOSBox.
If you want to install the game on Linux you either have to use Wine in order to run the Windows installer or you must install the game on a Windows PC and then transfer the game files into the Linux computer.
Once the game files have been installed on your Fedora Linux, you can use the preinstalled DOSBox in order to launch and play the game.
You should not worry about conflicts because the DOSBox version prepackaged with the game is for Windows, hence can't conflict with the linux version installed on your system.

Well, installing it on a Windows box and moving the files is Right Out because I run a Windows Free Household. I guess i'll just run the installer through Wine, test the program to make sure it works as installed and then, probably nuke the redundant Dosbox. Thanx!

Well, I thought it was solved, but I was wrong. If I launch Camageddon, I end up with a black screen until I switch to a different desktop and kill the program. Same thing with the Splat Pack. Any ideas?

If you have weak hardware running Linux:

I recommend using i386 build of the DOSBox on Linux, bc it outperforms x64 version by light-years.