I have a couple thoughts. First, it may not be as challenging to integrate with your existing CRM as you might think. There's some fairly simple ways to push information over via Zapier (a separate, paid service) or webhooks. Those still may not work but it's worth looking into if you haven't already.
Second, the primary CRM people connect to CallRail is Salesforce. There is a tight integration that allows significant customization if you decide to go down the SFDC route. Many customers also use Zoho CRM and connect to it via Zapier. You might want to see what CRMs Zapier connects to to see if our robust Zapier integration could be the best way to connect. I'm curious what other people would recommend as well so thanks for posting to the community!