The certificate has basically nothing to do with anything here. It's a bad sign that you've convinced yourself that's relevant to your problem.
The usual issue with Windows XP (specifically Internet Explorer on XP since e.g. Firefox works fine) is that it doesn't do Server Name Indication. Without SNI it isn't possible to implement virtual hosts for HTTPS sensibly. The only option is to assign a dedicated IPv4 address for each site to be hosted.
Cloudflare can do that for you, but they will charge. If it's important to have Windows XP Internet Explorer work, you should pay Cloudflare for one of the commercial options that they spell out as supporting Windows XP specifically.
Switching to Let's Encrypt won't make any difference to whether Windows XP works with your site _or_ to the SSL Labs score.