There is a another way, but it may not be ideal if you have to deal with 100s of Opportunities.
When an Opportunity is saved the data is posted to this link in your Infusionsoft Account: "/Opportunity/processOpportunity.jsp".
What I am thinking here is that you could have a script in the Browser Console Log to copy the Opportunity across to the second account by doing a Form Post to that Opportunity URL. Or, if that is not what you need is to adjust the script so that it updates the Opportunity Loss Reason.
It could become a bit complicated, and time consuming as you need to map the Opportunity IDs, etc. Might be quicker to manually do the changes.