Client certificate works in Firefox but not when used in Internet Explorer (or Edge)

With a certain web site, a certain client certificate works fine with Firefox, but not with IE or Edge (all on Windows 10, as both IE and Edge behave the same, I'll use the term "IE/Edge" from now on).

The error shown in IE/Edge is (as usual) pretty vague:

Can’t connect securely to this page

This might be because the site uses outdated or unsafe TLS security settings. If this keeps happening, try contacting the website’s owner.

IE/Edge using another client certificate (from different CA) connects without error to this same website.

Details:

• server is old (Weblogic 10.3.3, Java 1.6) : does not support sha2 certificates!
• nonworking certificate is from this CA (SIGEN-CA), old sha1 format
• the client certificate is old format (sha1)
• old root certificate was also sha1, but was later revoked and a new sha256 certificate was issued (with the same private key)

Any idea how to make IE/Edge also work in this scenario? (or if it can be solved by changes on the server side)