What does the reader look like? Can you post a picture? How the reader looks can tell us a lot, including whether it might be dual frequency or single frequency. If you shine your flashlight behind your credential, do you see one really thick antenna band that looks like a solid chunk of metal (LF prox) or do you see a thinner antenna ring or one with thicker wire and a few turns around the card(HF)?
When you scan the card with the Flipper, does it immediately recognize it as an HID card, if so, does it show it as H10301 format? H10301 is the format of the typical HID Proxcard which can be read, copied and emulated.
Additionally, what type of cards are you using to receive write commands? Blank rewritable T5577 cards are the best for copying any type of proxcard(including HID, but also including AWID or EMMarin)
So I’ve done some digging. Anyone, feel free to correct if this is wrong.
TLDR: Flipper supports 26 bit HID prox. Since your card isn’t 26 bit, it cannot write, emulate, etc.
I’ll drop some links to stuff for further reading, but what I’ve gathered is that Flipper can read other HID card formats, but can only copy, emulate, or write HID26 or 26-bit HID cards, commonly called HID ProxCard II. Your card is 35 bit Prox, which is likely part of HID Global’s Corporate 1000 program and likely uses a card format following the H5XXXX pattern. The specific format of an HID card that is in the Corp 1000 program varies with the corporation, but since the UID is 35, not 26 bits, flipper does not support the ability to write using this card. A proxmark3, if you have one, may be helpful in exploring this more in depth and enable you to clone the card.
As for what to do with this, there are no (to my knowledge) add-ons or other firmwares that have 33, 35, or 37 bit proxcard support. Unleashed is a great option, as it has some of the best extra tools, additional libraries and awesome community support (plug for the discord). Like I mentioned, you can also get a proxmark3 tool and use it to clone your card, though the command structure has a learning curve and is a command line based interface, which can be daunting if you are not used to it. There may be some other cheap cloners for prox out there, but I would be more hesitant to use them due to the fact that they may apply a password to your card which can affect future writes.