OK. I understand the client is just the browser and you have different domains in some of the URLs in the page. If the failure is due to the certificate a POST will not behave any different.
In your first post you said "Many users of my site use outdated computers". So far you have only described an older Macbook with a possible resolution that has not yet been tried.
If you have a wide variety of older clients you may need to switch to a different certificate authority. Lets Encrypt has the default "long chain" to allow use with older android clients but at the expense of certain other older clients. LE also offers a "short chain" which works for some but then not for older android. See this thread.
If you just have a handful of cases we could maybe direct you to options for them. Those clients may welcome such assistance as they are likely running into a variety of security related problems. For example, we just worked through a case with a very old android whose default browser failed but chrome worked. Sometimes it is easy (once you know!).