there are steel cables in that inner belt, should be ok, inflate and monitor for leaks, if it's keeping you awake at night, get a new replacement tyre, and donate the damaged one to people doing burnouts
I agree about technique, the process was super easy with the motionpro bead pro set until the bead problem surfaced, and once i shifted the portion of the tire 180 degrees from where i was working to be more towards the center of the rim, which again that was an easy adjustment, the rest of the job was painfully easy. Just took a few hours and the cost of a new Tractionator GPS from Motoz to learnIF using tire irons stiff wall tires will go better if you have help.
3 short tire irons and a couple clamps to like below to squeeze the opposite side of the tire into the drop center while working very small bites of the tire bead that is properly lubed.
If you are using excessive force and are hearing or seeing ripping and/or tearing STOP and regroup as you are doing it wrong. Even stiff walled tired are not that much harder to install verses soft walled sport tires. It's all about the technique.
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I'm using Motion Pro Bead pro's, all of the edges are rounded on them 😉Everyone’s got good comments but I have one more. Maybe look at getting a set of tire spoons that have soft edges vs using a pry bar with a hard 90 degree edge.
Windex use a great lube too, I've been using it for years.
Windex and dish detergent are very corrosive. If there's a little nick or scratch in the powder coat, these can cause a surprising amount of corrosion in the aluminum. I see this a lot on vintage bikes; you can often still smell the Windex or dish detergent, and the inside surfaces of the wheels are crumbly and look like they've been chewed by badgers.I've always used Ruglyde, which is made specifically for tire changing. It's around $20.00 for a gallon at places like NAPA. On a per ounce basis, it's about the same price as Windex.