You can do this in one turn provided you can Shove twice
SevenSidedDie's admonition that this doesn't have to be accomplished through established mechanics, and that improvised actions are appropriate and allowed, is the best answer. Mechanistically, this can also be accomplished through a grapple that then drags the opponent around you or through your space and into the fire.
For the sake of completeness, however, I would like to demonstrate how this can be accomplished through the mechanics of Shoving alone without, as the OP says, having to walk around the target: by first entering their space and then Shoving them twice toward the fire. This would take two separate Shoves and thus to achieve in one turn would require that you have Extra Attacks. You would also need to move 5 feet and expend 10 feet of movement. You would not provoke opportunity attacks (assuming enemies with 5' reach) unless there were enemies behind-diagonal to you (on either side of the fire).
First you enter their space
In order to accomplish what the OP desires - moving a creature in front of them to the space behind them, the PC must first enter the space of the opponent as part of their movement.
From PHB Chapter 9:
Moving Around other Creatures. You can move through a nonhostile creature's space. In contrast, you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you. Remember that another creature's space is difficult terrain for you. Whether a creature is a friend or an enemy, you can't willingly end your move in its space.
Notice what is prohibited by this rule - you cannot move through the space of hostile creature of similar size and you can't end your turn in their space. You are going to do neither. Instead, you are simply going to enter the space of the hostile creature and then move them out of it, and nothing in this rule prevents that, other than requiring you to have enough movement remaining so that you are not willingly ending your movement in the space (even after you have considered the extra cost of the difficult terrain) if you fail to move them out. Remember that you are allowed to break up your movement and attacks throughout your turn.
Once you have entered the opponent's space, you then figuratively adjust your positioning so that you are facing the opposite way as when you entered. Note that while the 'combat on a grid' system is very particular about how far you move (space to space) and which spaces you enter and leave, you are nonetheless free to change your facing any direction within a space, both during and out of your turn, as well as your positioning within the space itself (unless you are using the optional facing rules from DMG 252). Thus, this isn't even a 'step' as far as RAW are concerned, it is just something you can declare for clarity.
Then you make your first Shove attack
Instead of Making an Attack roll, you make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.
If your Shove is successful, you will have moved them to the space you occupied at the start of your turn. At this point you will have effectively traded places with them. They will have been moved from the same space you are currently in to the adjacent space by means of your Shove, having gone from 0 feet from you to 5 feet from you and thus fitting the standard English use of having moved them "5 feet away from you".
Finally, you Shove them again.
If your second Shove attack is successful, you will move them to the space behind where you were at the start of your turn, the space where the fire is, as desired by the OP.
Note again that if both your Shoves are unsuccessful, you must then move yourself out of your shared space with them, as you would otherwise be willingly ending your movement in their space, which is not allowed.
Narratively, this can be considered something like an elbow throw. While as much a Grapple as a Shove, it does demonstrate entering the opponent's space, turning 180 degrees, and throwing them beyond the space you formerly occupied.