Ulric,

Apologies for the delay, I realized after browsing down the tickets that yours slipped right through. (I know why - it's not a fast answer so I meant to circle back around and it slipped through).  If this ticket is still open for you and you need the answer, I can reply back with the exact database tables that I copy over from the live site to the staged site and how to do it so you don't lose database updates on those tables in the process.

Let me know,

Adam

Ulric,

Assuming there have been no new coupons or products created on the live site compared to the new J5 site, here are the database tables I usually copy over when transitioning from an active J2 site to the new one:
#__j2store_addresses
#__j2store_cartitems
#__j2store_carts
#__j2store_orderdiscounts
#__j2store_orderdownloads
#__j2store_orderfees
#__j2store_orderhistories
#__j2store_orderinfos
#__j2store_orderitemattributes
#__j2store_orderitems
#__j2store_orders
#__j2store_ordershippings
#__j2store_orderstatuses
#__j2store_ordertaxes
#__user_usergroup_map
#__users

To ensure that there are no database issues, what I do is export these tables as data only (not with structure) which would allow you to keep the Joomla 5 table's updated structure. What I do for each table:
Export the table (data only) (J3 site)
Create copy of the table (J5 site)
Empty the table so you only have the field titles with no data (not the copied table) (J5 Site)
Import the data only to the now empty table. 

Repeat this for each of the tables above.

If you find there are no difference in the number of items in the J3 and J5 tables then there is no reason to do the steps above (spot check to make sure the last item is the same for both)

If you are unable to import the data from the J3 site into the equivalent table, it's most likely due to a mismatch in the number of fields in both tables. (usually due to an older j2 plugin that could have added an additional field.  If this is the case and you want to keep the data on the new site, then install that plugin on the j5 site otherwise you would need to either add the extra field to that database or remove it from the live site. Importing them this way ensures you have no strange database issues on the new site.

I usually schedule a time later at night when the client and customers are usually not as active as you would need to put the J3 site in maintenance mode so you don't have new orders coming in after you have already exported them from the J3 site. You can test on the two user tables before hand to get comfortable, and once you put it in maintenance mode, it shouldn't take more than 1 hour to finish everything on the database side and swap the two sites out on the live site...

Let us know if you have any issues,

Adam

Hello Ulric,

apologies for the delay.

Can you create a ticket for Site 1, adding credentials as we cannot reproduce your issue on our test sites.

For Site 2, we can reproduce the error, therefore this is something we have to fix on our end.

Thank you!