Uh, that's a hard one. We have struggled with the general issue of upgrading and replacing the current system too. Users might register, content might be added, documents, ... all while we are working on the upgrade of the system.
I am afraid, there is no out of the box solution here. If your problem is limited to comments (it usually isn't) than you might get away with:
a) Write an export service that exports all comments (maybe after a specific date or something like that to get only the delta)
b) Write an import service that adds the missing comments
In my experience, the edit problem will affect more than just comments. Usually we do follow these steps:
1) Prepare the upgrade
Upgrade the database on a test system
Migrate the code, templates, configuration, ... on that dump
Write an upgrade guide (Change setting x, change ...)
We also often script some tasks
Please don't forget the you can export system settings an put them into the filesystem (This helps alot!)
2) Do a test upgrade
Basically, we do everything here what we would do during a real upgrade. If a major problem appears, we might even repeat this step. Usually some issues are found here, steps that were not documented, ...
3) Do the upgrade
On some date, often a weekend, we make the current production system readonly (e.g. by disabling the login). Sometimes we just add a big banner "Please do not edit, your edits will be lost ..." or even shut the system down and show some "Men at work" message. What we do, depends on the customer.
Then we upgrade a clone of the production system, deploy, apply the changes, run the scripts, do some tests and switch to the new server.
