I think so. Here's how you can try it:
Register your domain with CloudFlare, point your name servers at CloudFlare.
Set up your DNS as you normally would in CloudFlare, with A records for your domains and subdomains.
The domains you want CloudFlare to provide SSL for, make sure the "orange box" is turned on, on the DNS screen. This means CloudFlare will terminate the traffic.
The domains you want to use Let's Encrypt for you leave as "grey box" on the DNS screen. CloudFlare acts as a simple DNS server, has nothing to do with the connection between the browser and server, and you can do whatever you like regarding SSL.
You might like to consider CloudFlare full SSL. With that setup CloudFlare terminates your traffic, then re-encrypts to send to your server using the SSL cert on your server. This gives encryption from the browser to your web server. You can use any cert on your web server for this, Let's Encrypt or self signed. To use full (strict) mode is has to be a valid public certificate, which gives you a bit more assurance CloudFlare is connecting to your server.
All of this is available on the free CloudFlare accounts.
You've updated your question to explain a bit more about the "why": that you want to use expensive certificates for high volume sites.
IMHO, the free certificates from CloudFlare or Let's Encrypt are appropriate for many sites. I would suggest considering EV Certificates if you're dealing with financial information, or you need people to be sure it's your website, otherwise my opinion (which is knowing something about security but not being an expert) is other certificates are much the same. LE / CF are both trusted by most browsers.
You still need to be careful to secure your systems and resources, as SSL isn't everything. With CF you have to be especially careful to also encrypt from CF to your server. Then if the server is important you need to harden your server, patch your software, test for vulnerabilities, etc.