It would be interesting to know which takes precedence - I'd have thought a semi-final would be given priority over a fourth round tie so if they'd wanted Atkinson for a semi-final in the LC then someone else would have got the Monday FA Cup game. But who knows. I don't think Atkinson has been especially good this season anyway.
The FA Order of matches specifies that FA Cup is the most importance, so the League Cup being a FL competition would rank 3.
(b) The “Order of Precedence” of appointments, whether as a Referee or Assistant Referee shall be as follows:(1) The FA Challenge Cup Competition;
(2) The Premier League;
(3) The Football League;
(4) The FA Challenge Trophy Competition;
(5) The FA Challenge Vase Competition;
(6) Affiliated Association Cup Competitions*
(7) The Panel Leagues;
(

The FA Youth, FA County Youth Challenge Cup Competitions and FA Women’s Cup
4th round and above.
(9) Contributory Leagues (recognised divisions only), National League Systems Cup
and FA Sunday Cup
(10) FA Women’s Super League and Cup
(11) Supply Leagues (marking divisions only)
(12) FA Women’s Premier League and Cup
(13) Senior County Leagues
(14) Intermediate County League
(15) FA Women’s Cup prior to 4th round
(16) Women’s Combination League and Cup
(17) Women’s Regional League and Cup
(18) County Junior Leagues
(19) County Women’s Leagues and Cup
(20) All other competitions, including Youth Competitions