Can similar abilities used together in one action?

I have recently been tinkering around with a Fighter using Unarmed Fighting style.  This was a lot of fun.  But then suddenly there is a third-party class call Pugilist who just rolls right over all my testing with various feats and sub-classes.  While working out the details and differences I saw this interaction

Grapple feat has this amazing ability (along with a few others):

Punch and Grab. When you hit a creature with an Unarmed Strike as part of the Attack action on your turn, you can use both the Damage and the Grapple option. You can use this benefit only once per turn.

The new Pugilist base class has this gem at level 3.

Heavy Hitter. When you hit a creature with an Unarmed Strike, you can use both the Damage and your choice of the Grapple or Shove option.

Can this character use them both with the same Unarmed Strike?  Use Punch and Grab for the Grapple attempt and also the Heavy Hitter for the Shove.  Because this is crazy powerful.  Prior to this I was using a Battle Master and would use the Trip Attack Maneuver which resulted in a similar combo but at the cost of a scarce resource (a Superiority Die).  This Pugilist/Grappler combo is completely free.

Something else to keep in mind is that if the enemy is moved out of the range of your unarmed strikes, the grapple will be broken. This means that if you hit your enemy and successfully grapple them, a successful shove would push them out of your reach and break the grapple, rendering the entire tactic moot anyway. The only way this might be possible would be if you somehow have extra reach on your unarmed strikes (by something like the Warrior of the Elements Monk's Elemental Attunement feature).

Quote from SirBarnick >>

I have recently been tinkering around with a Fighter using Unarmed Fighting style.  This was a lot of fun.  But then suddenly there is a third-party class call Pugilist who just rolls right over all my testing with various feats and sub-classes.  While working out the details and differences I saw this interaction

Grapple feat has this amazing ability (along with a few others):

Punch and Grab. When you hit a creature with an Unarmed Strike as part of the Attack action on your turn, you can use both the Damage and the Grapple option. You can use this benefit only once per turn.

The new Pugilist base class has this gem at level 3.

Heavy Hitter. When you hit a creature with an Unarmed Strike, you can use both the Damage and your choice of the Grapple or Shove option.

Can this character use them both with the same Unarmed Strike?  Use Punch and Grab for the Grapple attempt and also the Heavy Hitter for the Shove.  Because this is crazy powerful.  Prior to this I was using a Battle Master and would use the Trip Attack Maneuver which resulted in a similar combo but at the cost of a scarce resource (a Superiority Die).  This Pugilist/Grappler combo is completely free.

 I don't see why you couldn't upon hitting a target with an Unarmed Strike this way, both damage it, Grapple it and Shove it Prone. Pushing it away would be counter-productive though if you only have ay reach of 5 feet.

But the essential point is that different game features can take effect upon the same trigger.

I'd apply the Simultaneous Effects rule in this scenario, so you can decide the order:

If two or more things happen at the same time on a turn, the person at the game table—player or DM—whose turn it is decides the order in which those things happen. For example, if two effects occur at the start of a player character’s turn, the player decides which of the effects happens first.

To me, the next example is similar:

Quote from TarodNet >>

[...] Now that you mention it, I guess the combo of Tavern Brawler and Grappler is interesting.

You could deal damage, then Push (Tavern Brawler), and finally Grab (Grappler feat), assuming your Unarmed Strike reach is 10 feet for some reason (e.g. a Monk Warrior of the Elements) 

Also keep in mind that whoever's turn it is decides the order of simultaneous effects. If you take an opportunity attack on another creature's turn, they could alter the sequence of events to be disadvantageous to you. For example, if you want to grapple a creature to reduce their speed to 0 (a flying creature might fall) and push them 5 feet away, they could choose to be pushed 5 feet away and out of range of your grapple, never being affected by it. On your turn, you would choose the order. It sounds like you just want to knock opponents prone, so it probably won't make difference.