Here is a way to get up and running with a new install of Firefox on Windows. Assuming you keep a backup copy of your custom profile folder, get it and keep it handy.

Install Firefox like always, say yes or no or whatever it asks— it doesn’t matter— you only want Fox to create a default profile with a name you can work with.

After that, exit Firefox and go find the profile it just created. For Windows at least (on 7 and 10), it should be here by default.

“C:\Users\username\AppData\Roaming\Mozilla\Firefox\Profiles”

The profile’s name will be formatted thus:

oei2zhc5.default

Within that profile folder’s location, copy/paste the profile folder from your previous install of Firefox.

...so that your folder and Ffx’s default will be listed together, like this

4tk7zhc5.default <—- Firefox default profile
myn7zhc5.default <—- your custom profile

Select the default profile and make like you’re going to rename it. Use F2. Then, while the name is highlighted, copy it with Ctrl + C. Then tap ESC. Now, either delete or rename the default profile, but for real this time.

Highlight your custom profile folder’s name with F2, and paste the default profile’s name with Ctrl+V, Enter.

Restart Ffx. You’re in. And you didn’t have to remake your settings from scratch.

ETA I am guessing that a similar method will work on Mac and Nix but I have little experience using Firefox on those systems.