There's no such thing as "semi-open source" It either is open source, or it isn't. The nature of open source makes it so that everything built upon it must be open source according to most of the open source licenses.

The idea you're proposing is very popular and used all the time by big successful companies. See Redhat, Atlassian, vTiger, SugarCRM, WordPress (Automattic). The key is offering value on top of the open source free/community package. The value for an enterprise would be your service and expertise in maximizing the software, professional support, or customizations for them, or even hosting it for them.

The upsides of doing it via open source is that the small companies that won't pay to license or service fees will customize some of it and hopefully contribute back to the software which will make it stronger for everybody. And as they grow they might eventually need your services or paid offerings.