Bakare committed a two-footed tackle on Dan Harding and was handed a straight red card after 25 minutes.
Welling goalkeeper Michael McEntegart denied Matt Tubbs and Yemi Odubade, before Jack Midson also hit the crossbar for the hosts.
The Spitfires are now seventh, three points off the play-off places.
Welling, who are 23rd and lie nine points adrift of safety, only had a 10-minute warm-up before the game after their team bus arrived late because of traffic.
Struggling Welling held on for a draw against play-off hopefuls Eastleigh, despite having Michael Bakare sent off in the first half.