External SD card capacity limit

According to Connect external storage devices to iPhone - Apple Support,

"An external storage device must have only a single data partition, and it must be formatted as APFS, APFS (encrypted), macOS Extended (HFS+), exFAT (FAT64), FAT32, or FAT."

I don't see any mention of a specific size limit, and Apple does say that exFAT is supported (which is crucial for handling cards with capacities of more than 32 GB).

You say that a 16 GB card works and a 256 GB card doesn't. If those cards are standards-compliant, the 16 GB card would be a SDHC card and the 256 GB card would be a SDXC one. I'd suggest checking the card reader. While a SDXC reader will read SD, SDHC, and SDXC cards, an old SDHC reader would only be designed to read SDHC and SD ones.

SD Association – Capacity (SD/SDHC/SDXC/SDUC)

Lightning ports supply very limited power

• When the reader + SD card tries to draw more than allowed:
  • iOS immediately cuts power
  • The warning appears (See the pic below). At times, this warning message may not appear
  • The accessory is disabled

No slow buildup, no countdown — it’s a hard stop.

• Use Apple’s Lightning to SD Card Camera Reader
• Or a reader with pass-through Lightning power
• Or move to iPhone 15+ (USB-C) — suddenly the power problem disappears like magic ✨