Based on now removed information the tag type of your source transponder is a MFC1C14_x which is a mifare classic ev1 chip with improved / hardened random number generator. It seems though that your proxmark3 was still able to determine access bits and make a dump of the protected memory in secured sectors. What’s not matching up here is the SAK, but there may also be other checks the readers are doing.
With the advent of magic chips out of china, it’s now possible and has been seen in the wild that readers will issue magic chip backdoor commands to the chip to see if it responds… and if it does then it will refuse to work. I’m unsure if your system is doing this, but it is a possibility.
One approach might be to obtain a magic gen1a and a magic Gen2 in card or key fob format and attempt to get the system to recognize those. The gen2 magic chip type does not use backdoor commands, it simply allows sector 0 to be written. The down side is that because there is no backdoor to get around all security settings, accidentally setting access bits incorrectly can result in a sector being locked out forever, just like any normal mifare chip. Some people have talked about ways to recover these types of sectors from a magic Gen2 chip but I don’t recall any actual proof.