It’s always hard to estimate costs, but from what you describe the cost could actually be pretty low. Indeed, this is the kind of case that Omeka is designed to make low-cost.
First off, if at this stage you aren’t looking for theme development, that drops the cost significantly, though depending on the grant options available it might be worth building in some of that time, just in case.
Second, if the hosting costs are already taken care of, then there’s not much there, as long what you have fits the standard needs of a LAMP stack. Reclaim Hosting, among others, works well, and has an easy installation process. Other hosting companies might require more time to install Omeka.
The points you list off are almost all available in Omeka out of the box. For anyone submitting, you’ll just need to add the Contribution plugin.
Since you have lots of content already, it sounds, then, like the greatest expense will be the person-time of setting up the site and moving the content over. That shouldn’t be underestimated, but I can’t give a real firm cost estimate. Factors are: time to set up the site itself; time to move content over; wages for the people doing that work.
Short version: it doesn’t sound like much, if any, developer (as in coding) time. It’ll just be setting up the site itself, then moving content over.