You can do this in one turn provided you can Shove twice
I agree that this could be accomplished through a grapple or an improvised action.
However, my reading of the "Shoving" and other movement rules would also allow this to be accomplished through Shoving alone, 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).
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. The standard English use of 'through' would imply that you are entering the space and then leaving out the other side. However, you are not going to do that. Instead, you are simply going to enter the space of the hostile creature and remain there, 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. 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.
Now 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. 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 will attempt to Shove them again. If 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 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]