So, I work in pretty much this exact area. The main issue is you're trying to sell something that customers don't buy - any integration, FHIR or HL7 or something, is only ever needed to get one product to talk to another. Usually a new one to talk to the old one; and the business selling the new product is the one in whose interest it is to get the integration working.

While it's always possible to take existing kit and do more/better automation, this kind of operational improvement tends to have little to no budget unless there's some specific project being undertaken (e.g. going to structured reporting). Generally, although people are often unhappy with internal software, the business will be operating as-is pretty comfortably. You'd need to deliver an awful lot of value to make it worthwhile paying you what you need.

I would probably look to better define your customer. You might have a better shout trying to approach ISVs who are selling new healthcare products, and asking them to use you as a delivery partner. Many ISVs have weak deployment teams and cannot deal with complex integrations, but they will be the ones with a customer and a contract. If you can speed up time to implementation, they get paid quicker and can do more.