Secondary Clustered DTC will not work when Primary goes offline - SQL Server Issues
I don't believe what you want to do is possible through Windows or SQL Server configuration. Per Microsoft's guidance here, SQL Server will choose an MSDTC instance when it starts and, if that MSDTC instance fails, distributed transactions will fail:

What happens if the MSDTC Resource that was picked at SQL Service startup fails?
If the MSDTC instance that is being used by a specific SQL Server Group fails, SQL Server does not automatically attempt to use the next available MSDTC Resource such as the default cluster instance or the local machine instance of MSDTC.
You would need to completely remove the failed instance of MSDTC from the SQL Server group to use another instance of MSDTC.
Likewise, if you create a mapping for SQL Server and the mapped instance of MSDTC fails, your distributed transactions will also fail. If you want  SQL Server to use a different instance of MSDTC, you must either add an instance of MSDTC  to the local cluster group of the SQL Server or delete the mapping.

This matches the behavior you're seeing; SQL Server successfully opens DTC transactions through DTC1 but doesn't automatically switch to using DTC2 if DTC1 fails.
In some cursory tests I found:

DTC transactions resumed working when the failed MSDTC instance recovered.

SQL Server was indeed able to automatically switch to using a different MSDTC resource without a restart after the MSDTC resource it was using was deleted (as opposed to just failed).


The tests I ran weren't comprehensive and there might be scenarios where these findings aren't true.
The more important point in the document is that you may not need to cluster MSDTC at all:

What will SQL do if I do not Cluster MSDTC in Windows 2008 and above?
If there is no MSDTC resource in the cluster then SQL Server will use the MSDTC service that is running locally on the node.
What is the Simple recommendation for MSDTC with Windows 2008 and later?
In Windows 2008 and later you either Create a clustered instance of the MSDTC resource for EVERY SQL Server instance /Group that requires its functionality or DO NOT CLUSTER MSDTC at all.

Based on this, you should be able to achieve pretty good high availability by relying on local MSDTC resources (in which case you might need to configure them to restart automatically on failure) or through adding a clustered MSDTC resource to each SQL Server cluster group if you prefer for some reason. Windows can handle restarting the service on failure, and SQL Server appears to successfully resume creating DTC transactions once the MSDTC service recovers.
If you're trying to protect against scenarios not covered here (e.g., what happens if the DTC resource fails and can't restart automatically?) you might be forced to deal with that manually or at the application layer.
It will be interesting to see if Microsoft provides you with different advice.