Here's what I've found after 17 years here in NC.
Tape and vet wrap, even electrical and duct, doesn't "survive" in our barn. When I do leave it in the barn, never fails that when I need it, it's melted into itself even when stored in a first aid kit in the tack area. And that sticky residue makes other stuff quite nasty. I make sure I always have some in the house and KNOW WHERE IT IS.
With switching to ponies, from Arabs, I don't use the first aid kit. So I have to make a point of checking it every so often (minimum of 6 months). My "biggies" were always getting injuries or having abscesses or getting sore. The ponies? NOT. Liquids will dry up, ointments go bad (even when you use gloves or implements). Electrolytes (even Gatorade) will get moisture in it. I no longer buy in large containers or in bulk for our first aid kit either. Every year when I get ready to have babies, I have to go get new Iodine and Nolvasan - Iodine turns thick and tarry, Nolvasan evaporates. That liquid stuff ... Cut Heal works great! But it doesn't last long since the applicator is stored right in the medication, LOL.
Also, since I don't have an air conditioned tack room or a refrigerator in the barn, I don't/can't leave drugs out there. Again, they stay in a container on my kitchen counter or in the fridge. It's pretty depressing to suddenly need some Bute and find that those white pills have turned crumbly and green! I have a "vet box" on top of my microwave that keeps meds for the dogs, cats, and ponies. There are flea treatment boxes, containers of whatever drugs being used (dry) etc.
Ponies and minis that are injured are often highly stressed. I used to keep a humane twitch in my first aid kit (don't currently have one). I used it on occasion so that I knew how to effectively apply it and the ponies knew what it was and didn't freak out more. The twitch often worked when I needed their minds taken off of an injury that was being worked on - namely legs being scrubbed and packed - rewrapped and needed them to stand on it not yank it away. If I have a helper, often used a chain dog collar drawn over the gums under the upper lip. Worked great. Again, most of ours had already been introduced to that before an injury occurred.
One thing I do have a problem with here - especially during our various pollen times thruout the year - is eyes. When they go on sale or if our local Optometrist will sell out of date to me, I get the big bottles of saline solution (my original vet would get cases of out of date saline for very low cost - some optometrists will only sell to a vet). Those do keep out in the barn - either in a tub or a cupboard. In a pinch, you can also make your own in a water/soda bottle - I had to when a mare I hauled two hours away to a drive developed a problem in the trailer with her eye and I had forgotten to take a bottle of saline solution with me (she regularly gets eye infections and was thought to be blind when I purchased her 5 1/2 yrs ago - she's not and I just keep checking her and she wears a fly mask more than any other pony we own).
You might want to swap out your high percentage Povidone for a lesser one (1% instead of 10%). I know that I've scalded several of mine with both 10 & 7 %... even when mixing it with something else for dilution.