Call up in place of Dylan Carlson, start against left-handed pitching, and utilize as a pinch-hit weapon off the bench.
You might be thinking "Doesn't that create the exact same issue the Cardinals are already in?". No, as Juan Yepez is better built for this kind of role than Carlson would be at the moment.
Yepez is a player who does not have major splits against right or left-handed pitching, so he becomes a very valuable bench bat in any game. In 53 at-bats for Memphis this year, Yepez has a .842 OPS and his bat is clearly Major League quality.
With Paul DeJong's hot start since returning from the IL, Brendan Donovan is a guy who can find more playing time in the outfield, so the Cardinals don't need to worry about not having quality enough defensive option to sub in for a Yepez or Burleson if needed.
When the Cardinals face left-handed pitching, Yepez can DH, two of DeJong, Nolan Gorman, and Tommy Edman can play in the middle infield, and the outfield can consist of Tyler O'Neill, Lars Nootbaar, and Brendan Donovan.
Yepez is a great bat to plug into the lineup whenever the Cardinals need an off-day for one of their corner infielders or outfielders and can pinch hit as a powerful bat in any spot. Yes, he would benefit from more playing time like Carlson, but Yepez doesn't need to "work on" his bat right now as Carlson does, so he can afford to play this reserve role right now.