Second instance of query designer for a single table in SSMS (Express)?
I don't believe there is a way to do this. SSMS seems to prevent you from opening more than one instance, and the only workaround I can think of is not to kick SSMS but rather just open a new instance of the entire application (as you've already guessed as well).

In any case, I strongly recommend you revert to your previous behavior - the query designer is full of bugs and your "edit data in place" feature actually means that the grid has to place unnecessary locks on the underlying table. On a concurrent system this can be disastrous - what kind of concurrency handling do you think is there? You should try to open two instances of SSMS and try to edit the same row in each copy. Fun fun!

Even though it is slightly more work to write, using proper DML (update/insert/delete) in a proper query window is a much better approach to ad-hoc data modification IMHO. And I have long been advocating to stay away from the SSMS visual designers in all their various forms. The team tried but some of the faulty code has been in there since Query Analyzer and it will never be fixed.