What is the quickest way to purge data?
Since you are deleting 90% of the rows, I'd recommend copying the rows you need to keep into a new table with the same structure, then use ALTER TABLE ... SWITCH to replace the existing table with the new table, then simply drop the old table.  See this Microsoft Docs page for the syntax.

A simple test-bed, without replication which shows the general principle:

First, we'll create a database for our test:

USE master;
IF (SELECT 1 FROM sys.databases d WHERE d.name = 'SwitchTest') IS NOT NULL
BEGIN
    ALTER DATABASE SwitchTest SET SINGLE_USER WITH ROLLBACK IMMEDIATE;
    DROP DATABASE SwitchTest;
END
CREATE DATABASE SwitchTest;
ALTER DATABASE SwitchTest SET RECOVERY FULL;
BACKUP DATABASE SwitchTest TO DISK = 'NUL:';
GO


Here, we create a couple of tables, with a trigger to move rows from table "A" to "B", approximating your setup.

USE SwitchTest;
GO
CREATE TABLE dbo.A
(
    i int NOT NULL 
        CONSTRAINT PK_A
        PRIMARY KEY CLUSTERED
        IDENTITY(1,1)
    , d varchar(300) NOT NULL
    , rowdate datetime NOT NULL
) ON [PRIMARY]
WITH (DATA_COMPRESSION = PAGE);

CREATE TABLE dbo.B
(
    i int NOT NULL 
        CONSTRAINT PK_B
        PRIMARY KEY CLUSTERED
    , d varchar(300) NOT NULL
    , rowdate datetime NOT NULL
) ON [PRIMARY]
WITH (DATA_COMPRESSION = PAGE);

GO
CREATE TRIGGER t_a
ON dbo.A
AFTER INSERT, UPDATE
AS
BEGIN
    SET NOCOUNT ON;
    DELETE
    FROM dbo.B
    FROM dbo.B b
        INNER JOIN deleted d ON b.i = d.i
    INSERT INTO dbo.B (i, d, rowdate)
    SELECT i.i
        , i.d
        , i.rowdate
    FROM inserted i;
END
GO


Here, we insert 1,000,000 rows into "A", and because of the trigger, those rows will also be inserted into "B".

;WITH src AS (
    SELECT i.n
    FROM (VALUES (0), (1), (2), (3), (4), (5), (6), (7), (8), (9))i(n)
)
INSERT INTO dbo.A (d, rowdate)
SELECT d = CRYPT_GEN_RANDOM(300), DATEADD(SECOND, s6.n + (s5.n * 100000) + (s4.n * 10000) + (s3.n * 1000) + (s2.n * 100) + (s1.n * 10), '2017-01-01T00:00:00.000')
FROM src s1
    CROSS JOIN src s2
    CROSS JOIN src s3
    CROSS JOIN src s4
    CROSS JOIN src s5
    CROSS JOIN src s6;


Clear the transaction log, to avoid running out of room.  DO NOT RUN this in production since it sends the transaction log data to the "NUL" device.

BACKUP LOG SwitchTest TO DISK = 'NUL:';
GO


This code creates a transaction to ensure none of the affected tables can be written to while we're migrating rows:

BEGIN TRANSACTION
EXEC sys.sp_getapplock @Resource = N'TableSwitcher', @LockMode = 'Exclusive', @LockOwner = 'Transaction', @LockTimeout = '1000', @DbPrincipal = N'dbo';
BEGIN TRY
    -- create a table to hold the rows we want to keep
    CREATE TABLE dbo.C
    (
        i int NOT NULL 
            CONSTRAINT PK_C
            PRIMARY KEY CLUSTERED
        , d varchar(300) NOT NULL
        , rowdate datetime NOT NULL
    ) ON [PRIMARY]
    WITH (DATA_COMPRESSION = PAGE);

    --copy the rows we want to keep into "C"
    INSERT INTO dbo.C (i, d, rowdate)
    SELECT b.i
        , b.d
        , b.rowdate
    FROM dbo.B
    WHERE b.rowdate &gt;= '2017-01-11T10:00:00';

    --truncate the entire "B" table
    TRUNCATE TABLE dbo.B;

    --"switch" table "C" into "B"
    ALTER TABLE dbo.C SWITCH TO dbo.B;

    --drop table "C", since we no longer need it
    DROP TABLE dbo.C;

    --shows the count of rows in "B" which were retained.
    SELECT COUNT(1)
    FROM dbo.B
    WHERE b.rowdate &gt;= '2017-01-11T10:00:00';

   --look for rows in "B" that should no longer exist.
    SELECT COUNT(1)
    FROM dbo.B
    WHERE b.rowdate &lt; '2017-01-11T10:00:00';

    --release the applock and commit the transaction
    EXEC sys.sp_releaseapplock @Resource = N'TableSwitcher', @LockOwner = 'Transaction', @DbPrincipal = N'dbo';
    COMMIT TRANSACTION;
END TRY
BEGIN CATCH
    DECLARE @message nvarchar(1000) = ERROR_MESSAGE();
    DECLARE @severity int = ERROR_SEVERITY();
    DECLARE @state int = ERROR_STATE();
    RAISERROR (@message, @severity, @state);
    EXEC sys.sp_releaseapplock @Resource = N'TableSwitcher', @LockOwner = 'Transaction', @DbPrincipal = N'dbo';
    ROLLBACK TRANSACTION;
END CATCH
GO


The sp_getapplock and sp_releaseapplock prevent multiple instances of this code running at the same time.  This would be helpful if you enable this code to be re-used through a GUI.

(Note that app locks are only effective if every process accessing the resource implements the same manual resource locking logic explicitly - there is no magic that "locks" the table in the same way that SQL Server automatically locks rows, pages, etc. during an insert/update operation.)

Now, we test the process of inserting rows into "A", to ensure they're inserted into "B" by the trigger.

INSERT INTO dbo.A (d, rowdate)
VALUES ('testRow', GETDATE());

SELECT *
FROM dbo.B
WHERE B.d = 'testRow'


+---------+---------+-------------------------+
|    i    |    d    |         rowdate         |
+---------+---------+-------------------------+
| 1000001 | testRow | 2018-04-13 03:49:53.343 |
+---------+---------+-------------------------+