The OP's questions seem to me to be based on certain errors.
He seems to regard a process as a single thing. Thus he takes the continuity of that process as equating to the persistence of a thing. Thus he faults the process for not bearing the mark of anicca. But as far as I know, the Buddha did not teach the 3 marks in this way. Simply, Theravada teaches the supporting condition for a moment of consciousness to be a previous moment of consciousness. Each consciousness moment arises and passes, thus bearing the mark of anicca, inconstancy. Thus not-self. That a process continues to function as long as the supporting conditions continue to arise does not mean constancy is to be found in the process. This is because the details of the process are always changing: one consciousness moment may follow another in an unbroken stream but a particular individual moment will be different than many of the ones before it. This is the mark of anicca, inconstancy. Because it is always changing it is unstable, thus stressful, thus not worthy of clinging to as a self.
Which brings me to the next error: the nature of anatta. The Buddha teaches anatta in a very specific way:
"Form, monks, is not self. If form were the self, this form would not lend itself to dis-ease. It would be possible [to say] with regard to form, 'Let this form be thus. Let this form not be thus.' But precisely because form is not self, form lends itself to dis-ease. And it is not possible [to say] with regard to form, 'Let this form be thus. Let this form not be thus.'"
Something is atta because it does not become ill or broken or troublesome; it is atta because it is controllable. The Buddha does not say (here or elsewhere as far as I know) that something is atta simply because it continues. Here's the other important thing he teaches about anatta:
"[It is not] fitting to regard what is unstable, stressful, subject to change as: 'This is mine. This is my self. This is what I am'."
Consider this: samsara is a process which continues. If merely continuing qualified it as a self then it would be fit to cling to. But it is precisely because the process of samsara is so unstable that it is stressful and therefore not worth clinging to. It is unstable because sometimes you've got pleasures and sometimes you've got pains and there's nothing you can do about that. In a similar way the consciousness process is unstable: sometimes your conscious of something nice, sometimes something not so nice. It is anicca, thus it is dukkha, thus it is anatta.
To call a process a "psuedo-self" merely because that process continues appears to me to be using one's own definition of "self" in place of the Buddha's.
When this arises, that arises. When this ceases, that ceases.
All compounded things are unstable.
[It is not] fitting to regard what is unstable, stressful, subject to change as: 'This is mine. This is my self. This is what I am'.
I hope this is helpful.