The Combat chapter of the PHB (p. 195-196) has this to say about shoving a creature:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your shove must be no more than one size larger than you, and it must be within your reach. 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.
Here, the word "shove" is defined as either pushing or knocking prone.
The Charger feat (PHB, p. 165) states:
When you use your action to Dash, you can use a bonus action to […] shove a creature.
If you move at least 10 feet in a straight line immediately before taking this bonus action, you […] push the target up to 10 feet away from you (if you chose to shove and you succeed).
Ignoring the "moving 10 feet" part, the bonus shove granted by the Charger feat seems likely to give the option to knock prone or push, as normal. The interesting part is when you do move at least 10 feet before the shove and succeed, as the text says that the target is pushed, but shoving is still, as its core, a choice between pushing and knocking prone.
If a Charger Dashes as his action, moves the 10 feet towards the target, then shoves as his bonus action but chooses to knock the target prone instead of pushing, would the target also be pushed up to 10 feet away?
It would make sense from a thematic standpoint (the charger ramming the enemy, sending them flying away and landing on their behind!), but from a mechanical standpoint, I'm not quite sure.
Best Answer
No; Charger removes the prone option.
It's right there in the text: After fulfilling the movement requirement, if you choose to shove and succeed, you push the target 10 feet away. That's the full effect of the enhanced shove, replacing the normal effects (including the option to knock prone), not in addition to them.
It wouldn't make sense to interpret the Charger "10 feet in a straight line" benefit as in addition to the normal results of a shove (i.e. a choice between knocking the target prone and pushing them back); that interpretation would imply that you could push a creature 5 feet and 10 feet at the same time by choosing the "push" effect. It seems clear that the intent of the feat was to allow you to push a creature 10 feet instead of the normal effect, and if that's the case, then it makes no sense to interpret it as letting you knock the target prone and push them simultaneously.
As a DM, I'd certainly allow a player to choose to knock the target prone with the charging attack, but they wouldn't gain any benefit beyond the normal shove effect of knocking prone. They'd just be choosing between knocking the target prone and pushing them back 10 feet, not both.