PHB comparisons:
Fighter 5 with two-weapon fighting fighting style and Dual Wielder
Main hand: rapier, attack action for two attacks, 1d8+5 each.
Off hand: rapier, bonus action for one attack, 1d8+5
Range 5 ft (35 ft with unobstructed movement)
+1 AC
Damage potential: 28 (3d8+15)
Fighter 5 with archery fighting style and Crossbow Expert
Main hand: hand crossbow, attack action for two attacks and bonus action for one more attack, 1d6+5 each.
Off hand: nothing
Range 30 ft (60 ft with unobstructed movement)
No melee disadvantage (doesn't need to leave melee combat to be effective)
+2 to hit
Damage potential: 25 (3d6+15)
With Sharpshooter:
Range 120 ft (150 ft with unobstructed movement)
No melee disadvantage (doesn't need to leave melee combat to be effective)
Damage potential: 65 (3d6+45)
Zapp's Redux II comparisons:
Fighter 5 with Crossbow Expert
Main hand: rapier, attack action for two attacks, 1d8+5 each.
Off hand: hand crossbow, bonus action for one more attack, 1d6+5.
Range 5 ft two attacks 30 ft one attack (35-60 ft with unobstructed movement)
No melee disadvantage (doesn't need to leave melee combat to be effective)
Damage potential: 27 (2d8+1d6+15)
Archery fighting style: +2 on one attack against targets with cover
Duelist fighting style: n/a
I would have chosen something else, such as Defense fighting style. This would match the +1 AC the dual rapier fighter gets.
In the end, you lose one measly point of damage in return for range on your off-hand attack - a great deal in my book. Not to mention how you get to play your cool archetype
In fact, I find the melee range to be a significant drawback, and so I have boosted the style by combining Defensive Duelist with Dual Wielder into a single feat:
Defensive duelist
Prerequisite: Dexterity 13 or higher
You master several fighting techniques, gaining the following benefits while you are
wielding a melee weapon in one hand and no shield:
• You gain a +1 bonus to AC provided you hold a light weapon or have one hand free.
• When another creature hits you with a melee attack, you can use your reaction to add
your proficiency bonus to your AC for that attack, potentially causing the attack to miss you.
• You can use two-weapon fighting even when the one-handed melee weapons you are
wielding aren’t light.
• You can draw or stow two one-handed weapons when you would normally be able to draw
or stow only one.
Note: pay attention to the precise language, it is carefully worded. (If I got it right, at least)
This means my dual rapier guy would look like this, and possibly not have two or even one rapiers!
Two rapiers: requires two-weapon fighting fighting style and loses out on the feat's +1 AC
One rapier: requires two-weapon fighting fighting style
Zero rapiers: you can pick defense fighting style for an additional +1 AC. Of course you could skip the feat entirely (since you're no longer using its arguably main benefit: using TWF with non-light weaponry).
Fighter 5 with two-weapon fighting fighting style and Defensive Duelist
Main hand: rapier, attack action for two attacks, 1d8+5 each.
Off hand: rapier, bonus action for one attack, 1d8+5
Range 5 ft (35 ft with unobstructed movement)
"Parry" (reaction to add proficiency to AC)
Damage potential: 28 (3d8+15)
Fighter 5 with two-weapon fighting fighting style and Defensive Duelist
Main hand: rapier, attack action for two attacks, 1d8+5 each.
Off hand: short sword, bonus action for one attack, 1d6+5
Range 5 ft (35 ft with unobstructed movement)
+1 AC
"Parry" (reaction to add proficiency to AC)
Damage potential: 27 (3d8+15)
Fighter 5 with defensive fighting style and Defensive Duelist
Main hand: short sword, attack action for two attacks, 1d6+5 each.
Off hand: short sword, bonus action for one attack, 1d6+5
Range 5 ft (35 ft with unobstructed movement)
+2 AC
"Parry" (reaction to add proficiency to AC)
Damage potential: 25 (3d8+15)