You are considering moving to SharePoint Online, and you already have a considerable investment in SharePoint 2010, here it is what I would suggest you in order to preserve you current assets and make a transition to the cloud smoothly:
1) Adopt a hybrid migration: it will not only preserve what you already have in place, but will allow you to gradually move your data to the cloud, while offering one single user experience through cloud, search will be the first visible property available on the cloud and the easiest piece of configuration to allow you audiences to find and discover information;
2) In the cloud you can access your data seamlessly, the data would still coming from your SharePoint On-Prem (assuming you were using BDC), it would still reach the cloud, and later you would start moving your configuration from On-Prem to the Cloud point to the On-Prem databases directly from SharePoint Online BCS, and eventually you would get a more robust recommendation to move the databases to themselves to Azure.
3) I would avoid a Big Bang migration: (creating everything from nothing in the Cloud), you would spend too much time in the process of creating your own migration scripts or going through the selection process of purchasing 3rd party tools, the other drawback is the user acceptance, SharePoint is powerful yet limited on the the development, restricting you of using server side solutions, while you are on a hybrid environment, your past investments using server side solution would still serve your users, giving you time to refactor solutions in the SharePoint Framework (SPFx) and adapt things into a client-side and match in Azure jobs what can't fit in SPFx.
