How to replace a single table with Identity Primary Key column from a database backup?
Based on the information you have provided in your question, I think you will need to restore a good backup (with the missing rows for Incidents) into a test database.  You'll need to insert into the real current Incidents table the missing rows from the backup version of the Incidents table (making sure you properly use the SET IDENTITY_INSERT command).  Using SET IDENTITY_INSERT ON allows explicit values to be inserted into the identity column of a table thus allowing you to insert the deleted rows.  Make sure you issue the SET IDENTITY_INSERT OFF after you have completed your inserts.  You'll need to do the exact same comparison and insert for the children tables for each of the parent rows you are restoring since the children were automatically deleted when the parent rows were deleted (due to cascade delete FK's).

After you have addressed all children tables, you should be able to add the FK's back again with CHECK.

You might want to prototype this on a test server.

--Demo IDENTITY_INSERT
--Create test table
IF OBJECT_ID('dbo.TestTable') IS NOT NULL
    DROP TABLE dbo.TestTable
GO

CREATE TABLE [dbo].[TestTable] (
    TestId INT identity(1, 1) NOT NULL
    ,TestColumn VARCHAR(5)
    ) ON [PRIMARY]
GO

--insert a few test rows
INSERT INTO TestTable (TestColumn)
VALUES ('one'),('Two')

--select the two rows - the id values will be 1 and 2
SELECT * FROM TestTable

--delete the two rows
DELETE FROM testtable

--set identity_insert on to allow you to explicitly insert id values
SET IDENTITY_INSERT dbo.TestTable ON

--insert id 1 and 2
INSERT INTO TestTable (TestId,TestColumn)
VALUES (1,'one'),(2,'Two')

--set identity_insert off
SET IDENTITY_INSERT dbo.TestTable OFF

--select the current rows - should be id 1 and 2
SELECT * FROM TestTable