SSMS 2008 Object exploer not showing

In my case, I was able to solve the problem by reducing the size of the transaction in the stored procedure, ala the comment provided by @Aaron Bertrand. The entire stored procedure was wrapped in one begin/end transaction. As the stored procedure progressed, my management studio instance would lock up. Reducing the transaction to just the critical inserts/updates, and adding some error handling, eliminated the problems I was having with SSMS.

SQL Management Studio (versions prior to 2016)

Unfortunately, there are some caveats that make using Application Intent in SQL Management Studio somewhat painful:

To connect manually with ReadOnly intent, after bringing up the Connect to Server dialog from the Object Explorer, users must remember to:

1. Click Options >>.
2. Go to the Additional Connection Parameters tab.
3. Enter the additional parameter as ApplicationIntent=ReadOnly;
4. (Note: Users must not click the Options << button after entering the Additional Connection Parameters or the parameters will be lost.)
5. Click Connect.
6. Always launch query windows by right-clicking on the desired database in the Object Explorer view and choosing New Query to avoid running into the #3 caveat below.

The caveats that apply are as follows:

1. Although you can get SQL Management Studio to connect with Read Only Intent, it does not store the Additional Connection Parameters when a connection is added to Registered Servers.
2. Behavior when hand editing the locally registered servers in the RegSrvr.xml file to add the Application Intent is extremely inconsistent and will be overwritten any time a change is made through the GUI making this workaround unreliable.
3. The Always On database must be selected before the query window is opened; otherwise, the connection gets routed to the primary server. If you attempt to select the database using the query window's drop down after the query window has already been opened to a non-Always On database, you will get an error dialog. If you try to change the database to an Always On database with a USE statement after the query window has already been opened to a non-Always On database, the results look like this when you attempt to execute the SQL query:

      Msg 979, Level 14, State 1, Line 1
      The target database ('AlwaysOnDatabase') is in an availability group 
      and currently does not allow read only connections. For more 
      information about application intent, see SQL Server Books Online.

SQL Management Studio (versions 2016 or later)

SQL Server Management Studio 2016 or higher can connect with Read Only Application Intent (using the same 6 steps as prior versions) and it does store the Additional Connection Parameters. There are still some caveats:

1. The Object Explorer view will not enumerate any of the tables or other objects in the AlwaysOn databases. Attempting to expand them results in an access denied message.
2. You cannot have a Read Only and a non-Read Only connection to the same listener open at the same time.
3. Intellisense for object names in the database does not work. (Oddly enough, Objects get enumerated just fine in the Query Designer that you can launch using Design Query in Editor... from the right-click menu.)
4. Caveat 3 from the prior versions' caveats still applies.

Third-Party Products

LinqPad stores the whole connection string including Application Intent and the database when you save a connection and therefore might be a viable option for performing Read Only queries against Always On databases.