Theoretically, this could be done, but in practice, you'll probably need to find somewhere else to store that information. Each end of the connection is allowed to choose any arbitrary number it wants, but using anything other than a completely random number leaves a gaping vulnerability to certain types of attacks. This might be acceptable to you, if the machine in question is on a tightly controlled network, but because of how bad an idea this is, you will essentially need to build your own TCP layer; operating systems generally won't expose an API to shoot yourself in the foot like this.
And even then, you'll need to do this at both ends, since each end of the connection chooses its Sequence Number independently. And you'd need to make sure rollover is handled properly. If you really and truly insist on this, then that's fine, but the machines that you do it on will have issues properly talking TCP with anything else due to the issue of rollover, and you're going to need to rewrite a lot. You'll almost certainly be better served trying to find somewhere else to store that session information.