Late to the party but I had to jump in because I have been in your exact shoes with the 5D Mark III. I remember shooting a high-stakes event years ago where I missed the first kiss shot because my buffer was hanging... it is a gut-wrenching feeling. Over the years, I have realized that while the body is a total tank, you really have to treat it right with the peripherals or it will bite you. Since someone already mentioned the SD card bottleneck, I will tell you what else saved my skin. First off, you need absolute reliability. Since you have a decent budget, dont skimp on power or lighting. Here is what I would grab to make that setup actually feel professional:

• Canon LP-E6NH Lithium-Ion Battery
• Lexar Professional 1066x 128GB CompactFlash Card
• BlackRapid Sport X Breathable Camera Strap
• Canon Speedlite 600EX II-RT Flash These bits of gear are what I used to finally stop worrying about my kit. The Lexar cards are super snappy for that specific CF slot and I have never had one fail. Also, that BlackRapid strap is a lifesaver. It puts the weight on your shoulder instead of your neck, which is huge for a heavy 5D setup. If you can, grab at least three of those OEM batteries. Running out of juice mid-ceremony is a nightmare I wouldnt wish on anyone. You got this tho, that camera is still a legend when it is rigged up right.

The 5D Mark III has a massive bottleneck with its SD slot because it doesnt support UHS speeds. Thats why your buffer is locking up during the ceremony. You honestly need to move your primary recording to a fast CompactFlash card instead. I suggest these for a reliable pro workflow:

• Storage: Use the SanDisk Extreme Pro 128GB CF 160MB/s. It clears the buffer way faster than any SD card can in that body.
• Power: Grab the Canon BG-E11 Battery Grip. Dont go third-party here tho... they often lose connection or the buttons feel mushy mid-shoot.
• Charging: The Watson Duo LCD Charger for LP-E6N is great for overnight prep since it handles two packs at once with a clear readout.
• Ergonomics: Switch to the Peak Design Slide Camera Strap. It distributes the heavy 5D weight across your shoulder instead of your neck. This setup is gonna fix those speed issues and keep you running through the reception.

Finally someone says it. Ive been thinking this for a while but wasnt sure.

I've been using the 5D III since it was the industry standard, and honestly, you need to be really cautious about what you're putting in those slots. To fix that buffer lag, I would suggest moving exclusively to high-speed CF cards from a reputable brand like Delkin or ProGrade. Basically, dont even bother with the SD slot for anything other than a low-res backup because it'll just slow you down and cause that lockup you're seeing. On the power side, you really should stick to the official Canon brand batteries. I've seen third-party ones report 50% charge and then just die instantly during a reception... it's a huge risk for a wedding and just wont be worth the savings if you miss a shot. For the neck pain, you might want to consider a leather harness system from a brand like HoldFast. It moves the weight off your neck and onto your shoulders, which basically saves your back during those long days in Chicago. Just make sure you get everything from a legit dealer so you dont end up with a counterfeit card that fails on you when the pressure is on.