Hey there! Welcome to the backup LED lights club. I just did the same thing last year on my 2001. Although I had to install 1 resistor as the backup camera got fooled by the LED to turn on even though I was not in reverse.
I've had LED back up bulbs in my 2014 since ... probably 2014. I love them. I had bought some el-cheapo Amazon units (for a few bucks each) and replaced them every few months and then when @Diode Dynamics had a sale, I bought theirs and have never had an issue.
Also got their side marker, brake, turn signal and other bulbs. If you do go that route, it can get expensive and you'll also need to find an LED resistor blinker module (not sure if DD's tap/flash module fits older Foz) but you get much brighter lighting all around.
I'm pondering of adding some eagle eye switchback reverse/turn signal LED's to the lower license plate holes for even better ground visibility for me, other drivers and pedestrians....what do you guys think?