One option, as Jachym Michal pointed out, is to adjust the color of the checkerboard to match the one from the background. This answer is meant to provide more detailed instructions in case somebody else wants to try this approach but doesn't find the theme settings. Please note that this will adjust the look of the transparent checkerboard pattern in all areas of Blender.
The background color for the 3D View in the default theme is
#393939 (gamma corrected). This value can be found in the theme preferences (Edit > Preferences > Themes) under the name Gradient High/Off (3D Viewport > Theme Space > Gradient Colors).
The color settings for the checkboard pattern can be found in User Interface > Transparent Checkerboard and are called Primary Color and Secondary Color. Setting both of these to
#393939 will make it indistinguishable from the regular background.