and then edit the name of the new tag back to the old name.
I don't think you can do this step while online to a tag currently in use.
Just to summarize (as I see it) - Current Status:
Alias -> Target
BHW_03_Running -> RIO2102_S1_MCMR.DATA.ReadData[44].2
Desired
BHW_03_Running -> RIO2102_S1_MCMR.DATA.ReadData[44].3
Step 1 - Create
BHW_03_Running_New -> RIO2102_S1_MCMR.DATA.ReadData[44].2
Step 2 - Replace
Replace each BHW_03_Running with BHW_03_Running_New
This may very well be an individual 'Find / Online Edit' operation. I don't think the automated 'Find/Replace' is available online. Though the 'cross-reference' on that individual tag gives you a handy table to choose from. Come back to the table and choose 'refresh' after each edit. During this process the logic is still dealing with the status of the same point, just using different aliases.
Step 3 - Modify
Since it's now not being used you may be able to directly change the alias target for BHW_03_Running or you may have to delete and re-create it. While deleted your display system will complain.
Step 4 - Replace again
Replace each BHW_03_Running_New with BHW_03_Running
Again the 'cross reference' should make this faster. But be aware that while doing this replacement your code will be dealing with the status of two different points. Make sure this won't have bad consequences.
When done you can delete BHW_03_Running_New.
Good luck.