I don't think that there is a good answer for this problem, except starting over from scratch on one of my computers, but I shall ask regardless. Originally, I didn't intend to install my email client (The Bat) on my secondary computer, but due to the fact that the other computer went down for a few days, I did anyway.
Now, with both machines running, instead of both machines receiving the same messages, creating a redundant record in each, the computers split the messages, depending on which checks Gmail first. The email is left on the Gmail account online, but when the second machine checks, it is though Gmail sees both machines as the same one, and doesn't send the messages again.
I believe this is due to the fact that I copied the application data from one machine to the other, thus something in that data has an ID mark somewhere that masquerades each machine as being the same. I have no idea of where to look, and even if I did, no idea of how to edit it, without causing other problems. Chances are, I will just reset the secondary client to not check email at all, so that all messages come to the primary one, but if I need to access an old message when the primary is down for some reason, the only way that I could do so is online at Gmail, which I dislike doing, because I rarely label and archive the account, and have a very large pool of messages in the inbox.
I don't really expect a usable answer for this, but I'm hoping that someone might know a trick.