To the best of my knowledge, there is no way to solve this, at least without severely breaking signature checking. Other mail clients behave similar to Outlook, although I think that there is no good reason for this behavior.
I have looked over the GPO settings, but I found none that would fix this problem and leave everything else intact. There is a setting "Promote Level 2 errors as errors, not warnings" though; its description says that enabling the setting degrades some problems down to warnings that would otherwise be errors (in my opinion this description is the opposite of what the name suggests). The expiration issue is not in the example list of problems, but it might still work, as the problems there are much worse (like not finding the signing certificate at all). I didn't try this because of the vast extent of this setting, but it might be an option if you are really desperate ;-).
Outlook 2010 and 2013 also show this behavior, it seems to violate the RFCs, but Microsoft currently does not work on the issue: https://social.technet.microsoft.com/Forums/office/en-US/ec808b17-ee00-4717-9fc1-f877085dc34c/outlook-2010-signed-messages-are-falsely-marked-as-having-an-invalid-signature?forum=outlook
I can confirm that this is still reproducible in Outlook 2016.
In my tests, Thunderbird behaves identically to Outlook and even falsely says that the cause for the problem was "The certificate used to sign the message was issued by a certificate authority that you do not trust for issuing this kind of certificate." Thus, this seems to be a general problem.
See also this similar question on Information Security: https://security.stackexchange.com/questions/52254/reading-older-mails-signed-with-a-certificate-that-has-meanwhile-expired
Get a new signing certificate long before your old one expires, like 6 months, and start using only the new certificate. This has the following benefit: When the old certificate expires, mails with seemingly invalid signatures are at least 6 months old. Your recipients hopefully only seldomly re-read mails that are 6 months old and older and therefore are less likely to run into the problem.