To be a reversible process, information must be preserved - by definition: you cannot reverse something, i.e. put things back to the way they were before it happened, if you've lost some of the information describing how things used to be.
As the demon decreases the entropy, it has to keep tabs on which chamber each molecule is in so it doesn't let one go back the other way. Thus it needs to make a new one-bit entry in its memory (chamber "A" or chamber "B") for that molecule-passage event. But its memory is necessarily finite, so there will come a time there is no more space for any new records. Once the memory has now exhausted all bits, there is no other place to put the new information. Reversibility would mean you can move information around in memory (i.e. shuffle around the records of what molecules went where), but you can't put anything new to make for this new molecule. To put something new, you have to throw information out of memory - into the Universe, which means it becomes lost, and the process is irreversible as you now no longer have a bit of information you need to put things (the demon and boxes) back to the way they were before. Or, you have to ignore any new molecules, meaning either the demon will not be manning the door anymore or it will have to start doing something stupid like start flipping it back and forth at random, and entropy will cease to decrease.
You thus cannot "write over it reversibly" because the information that was there before that write has to still be there, somewhere else in the system, for you to be able to reverse the write. With a finite memory capacity that is smaller than the capacity of the thermodynamic system, that is not possible. Eventually, whatever "auxiliary" store you may be using, even though you did not call it "memory", will fill up, and then you are left with either cessation of the demon's job or with irreversibility (ejection of information into the outside Universe) as the only two possible choices.
