Here's a start . . .
*Step 1* - Set up a group to oversee the implementation and draw up a draft budget.
*Step 2* - Engage a dedicated 3rd-party consulting team to review the current systems and assess needs - they should speak with all project stakeholders including (and especially) end-users.
*Step 3* - Engage a separate systems integration team to scope a solution, directly addressing the criteria listed in the review.
*Step 4* - Create a firm budget and timeline.
*Step 5* - Get you legal team to draw up a contract. Make sure it is detailed, listing specific processes, outcomes, timelines, budgets and appropriate penalties for failing to meet any of them. Set mandatory review points, both by schedule (to assess progress) and at the completion of any major function/component. Reviews should include representatives from all stakeholders.
*Step 6 * - Review, test and communicate continually.
Or, to simplify: plan properly, budget properly, manage properly and review properly.
Let's go one simpler: treat it as if it was your own money.
Engaging a single entity to do it all and just letting them get on with it has the potential to be quicker and cheaper if it all works out well. Unfortunately, it is almost assured to work out quite badly indeed, costing far more in overruns and 'fixes'.
Take the time and put in the work to plan it out properly and you will almost always come out ahead with a more functional system that better meets your needs, delivered far closer to on time and in budget.