I quickly looked over this post to get the facts and see what has already been established.
From my understanding your keycard is an EM410 with the ID 001A338701
I am guessing that the T5577 data you got was from the NExT? If so, that is good and to be expected.
From the output you gave of
lf tune Ive got a feeling you are running an older version of the client.
Could you share the output of the command
hw version please?
The tricky part, from my understanding, is trying to get placement on the proxmark LF antenna so that it couples well and issuing the command on the client to read said implant.
You can issue multiple proxmark commands in the one line to be executed sequentially by using a semicolon between each command. For example, if you wanted to run 3 low frequency searches back to back you can issue
lf search; lf search; lf search on one line and press enter.
Once you have been able to get used to positioning your hand and the proxmark so that it couples well, you can move on to writing new data to it. The good news is that copying an EM410 card to a T5577 is fairly straightforward. But I would like to know the output of the version command above plus the assurance that you are getting better reads before suggesting a write.