As I mentioned, I was on version 10.2 when I did this.  As far as I know, none of these database tables or fields were updated until version 11 Update 2 (at that point, the user_id field was changed to user_group_id or something like that), but I'm not 100% positive that no other changes were made.

Here's the queries I ran.  Please proceed carefully.

--Creating Variables for Retrieving Username, ID, and SID
DECLARE @OldUserName VARCHAR(100), @OldUserID INT, @OldUserSID VARCHAR(100), @OldUserDisplayName VARCHAR(100), @OldUserEmail VARCHAR(100)
DECLARE @NewUserName VARCHAR(100), @NewUserID INT, @NewUserSID VARCHAR(100), @NewUserDisplayName VARCHAR(100), @NewUserEmail VARCHAR(100)
SET @OldUserName = 'Matthew'        --Manually entry the value here for the old username.
SET @NewUserName = 'DOMAIN\matthew'   --Manually entry the value here for the new username.

--Retrieving Details of Old User
SELECT
  @OldUserID = [user_id],
  @OldUserSID = [sid],
  @OldUserDisplayName = [displayname],
  @OldUserEmail = [email]
FROM [LFForms].[dbo].[cf_users]
WHERE [username] = @OldUserName;

--Retrieving Details of New User
SELECT
  @NewUserID = [user_id],
  @NewUserSID = [sid],
  @NewUserDisplayName = [displayname],
  @NewUserEmail = [email]
FROM [LFForms].[dbo].[cf_users]
WHERE [username] = @NewUserName;

--Displaying Details of Old and New User
SELECT
  @OldUserName AS old_username,
  @OldUserID AS old_user_id,
  @OldUserSID AS old_user_sid,
  @OldUserDisplayName AS old_displayname,
  @OldUserEmail AS old_email;
SELECT
  @NewUserName AS new_username,
  @NewUserID AS new_user_id,
  @NewUserSID AS new_user_sid,
  @NewUserDisplayName AS new_displayname,
  @NewUserEmail AS new_email;

--If the new userid has never been used, it's profile doesn't exist yet, in which case
--it needs to be added to the database.
INSERT INTO [LFForms].[dbo].[cf_user_profile] ([user_id],[sid])
SELECT
  u.[user_id], 
  u.[sid]
FROM [LFForms].[dbo].[cf_users] AS u
LEFT JOIN [LFForms].[dbo].[cf_user_profile] AS p ON p.user_id = u.user_id
WHERE u.[user_id] = @NewUserID
  AND p.sid IS NULL;

--Update the new user profile to copy all the values from the old user profile.
--This ensures that things like signature, favorite forms, column selections will remain for the user through the switch.
UPDATE [LFForms].[dbo].[cf_user_profile]
SET [fav_bp_list] = (SELECT [fav_bp_list] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [my_task_col_width] = (SELECT [my_task_col_width] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [my_history_col_width] = (SELECT [my_history_col_width] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [show_pm_palette] = (SELECT [show_pm_palette] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [tz_setting] = (SELECT [tz_setting] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [fav_view_list] = (SELECT [fav_view_list] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [home_page] = (SELECT [home_page] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [my_task_hash] = (SELECT [my_task_hash] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [my_instance_hash] = (SELECT [my_instance_hash] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [manage_hash] = (SELECT [manage_hash] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [my_task_groupby] = (SELECT [my_task_groupby] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [manage_sortby] = (SELECT [manage_sortby] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [signature_type] = (SELECT [signature_type] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [type_name] = (SELECT [type_name] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [type_font] = (SELECT [type_font] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [data_url] = (SELECT [data_url] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [thumbnail_data_url] = (SELECT [thumbnail_data_url] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [notification_options] = (SELECT [notification_options] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [process_metrics_info] = (SELECT [process_metrics_info] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [inbox_settings_json] = (SELECT [inbox_settings_json] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [approval_settings_json] = (SELECT [approval_settings_json] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID),
    [mobile_inbox_settings] = (SELECT [mobile_inbox_settings] FROM [LFForms].[dbo].[cf_user_profile] WHERE [user_id] = @OldUserID)
WHERE [user_id] = @NewUserID;

--Update User Snapshots to list the new user details for all Snapshots.
--This will ensure the forms assigned to the old user, and all form history showing the old user,
--will automatically be switched to the new user.
UPDATE [LFForms].[dbo].[cf_user_snapshot]
SET [user_id] = @NewUserID,
    [username] = @NewUserName,
    [email] = @NewUserEmail,
    [displayname] = @NewUserDisplayName,
    [sid] = @NewUserSID
FROM [LFForms].[dbo].[cf_user_snapshot]
WHERE [sid] = @OldUserSID;

--Update Any Other User Snapshots for the new user, to ensure none are missing email or displayname.
UPDATE [LFForms].[dbo].[cf_user_snapshot]
SET [email] = @NewUserEmail,
    [displayname] = @NewUserDisplayName
FROM [LFForms].[dbo].[cf_user_snapshot]
WHERE [user_id] = @NewUserID
  AND ([email] IS NULL OR [displayname] IS NULL OR [email] = '' OR [displayname] = '');

--Update Business Process roles from the old user to the new user.
--These are the access rights to edit or start particular processes.
UPDATE [LFForms].[dbo].[cf_users_in_role]
SET [user_id] = @NewUserID
WHERE [user_id] = @OldUserID;

--Update User Dashboards from the old user to the new user.
--These are part of the reports functionality.
UPDATE [LFForms].[dbo].[dashboards_by_user]
SET [user_id] = @NewUserID
WHERE [user_id] = @OldUserID;

--Update Team Memberships
UPDATE [LFForms].[dbo].[team_members]
SET [user_id] = @NewUserID
WHERE [user_id] = @OldUserID;

--Update permissions granted to reports to move them from the old user to the new user.
UPDATE [LFForms].[dbo].[cf_view_security]
SET [user_group_id] = @NewUserID
WHERE [user_group_id] = @OldUserID;

--Update permissions granted to dashboards to move them from the old user to the new user.
UPDATE [LFForms].[dbo].[dashboard_security]
SET [user_group_id] = @NewUserID
WHERE [user_group_id] = @OldUserID;

--Update fields on in-progress instances where username is a field value.
UPDATE [LFForms].[dbo].[cf_bp_data]
SET [value] = @NewUserName
FROM [LFForms].[dbo].[cf_bp_main_instances] AS i
JOIN [LFForms].[dbo].[cf_bp_worker_instances] AS wi ON wi.[bp_instance_id] = i.[bp_instance_id]
JOIN [LFForms].[dbo].[cf_bp_worker_instance_history] AS history ON wi.[instance_id] = history.[instance_id]
JOIN [LFForms].[dbo].[cf_bp_steps] AS step ON wi.[current_step_id] = step.[step_id] AND wi.[current_process_id] = step.[process_id]
JOIN [LFForms].[dbo].[cf_bp_data] AS d ON d.[submission_id] = history.[submission_id]
JOIN [LFForms].[dbo].[members] AS m ON m.[id] = d.[member_path]
WHERE i.[status] = 1
  AND history.[submission_id] IS NOT NULL
  AND d.[value] = @OldUserName
  AND d.[member_path] NOT LIKE '%.%'

--Report Business Process steps that reference the old username.
SELECT 
  bp.[name] AS process_name,
  step.[step_type],
  step.[name] AS step_name,
  'This business process step has a hard-coded reference to the old username, which needs to be updated manually.' AS instructions
FROM [LFForms].[dbo].[cf_bp_steps] AS step
LEFT JOIN [LFForms].[dbo].[cf_bp_processes] AS bpp ON bpp.[process_id] = step.[process_id]
LEFT JOIN [LFForms].[dbo].[cf_business_processes] AS bp ON bp.[bp_id] = bpp.[bp_id]
WHERE [definition_json] LIKE CONCAT('%"', @OldUserName, '"%')
  AND step.[is_deleted] = 0
  AND bp.[is_deleted] = 0;

This was also after I'd run the Synchonization in Forms so that both the old user and new users already existed in the LFForms database.

Thanks Matthew!