The #1 takeaway I have from my time watching small companies try to integrate into big EMRs in surgery areas is to find a physician champion. Convince one physician who will make it happen. Periop and procedural areas are run like an assembly line and are more rigidly opposed to change in their workflows and processes than a lot of the rest of the hospital. As a small company, your biggest obstacle will be to convince the hospital IT shop that you're worth the trouble and security risk. The best way to do that is to convince the customer of the hospital IT department - i.e. the physician champion.
A company I think has a great strategy for this is Redox, have a look at their strategy kit: https://www.redoxengine.com/blog/integration-strategy-kit They've got a great blog too: https://www.redoxengine.com/blog
I'd be thrilled to have a chat with you about your implementation strategy.