With your houserule, the DPR is huge
Natural Weapons are not Unarmed Strikes, making them so is a very unofficial, very unbalanced houserule.
Level selection
The best damage available in beast form is 24(4d8+6, Triceratops Gore) at +9 to hit, but you need 15 levels of Druid for it. To get the Extra Attack, you have to have 5 levels of Monk.
This is all of your 20 levels.
Starting Ability scores
for Human Variant with the Resilient(Con) for Concentration:
Str: 8
Dex: 16
Con: 14
Int: 10
Wis: 16
Cha: 10
ASI Monk 4: Sentinel, for out of turn DPR
ASI Druid 4: +2 Wis, for AC
ASI Druid 8: +2 Dex, for Flurry of Blows
ASI Druid 12: +2 Dex
Calculation
Assume AC 20 for the enemy, it is not unreasonable on level 20.
Official DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+11 to hit, 2 x (3.5+5) (Flurry of Blows) = 0.6 x 2 x 3.5 = 10.2
Alltoghether it is 34.2 vs AC20.
This is a very sudden jump however, at level 19 you either do not have Triceratops, or Extra Attack. Without Extra Attack you do 12 + 10.2 DPR, below a 17th level Monk.
Houserule DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+9 to hit, 2 x 24 (Gore as FoB) = 0.5 x 2 x 24 = 24
Alltoghether it is 48 vs AC20.
Extra Attack makes a much smaller difference now, as the FoB does the same damage as attacks done with your action. This is very unbalanced.
Compare it to 17 levels of Monk:
+11 to hit, 2 x (1d10+5) (basic attack) = 0.6 x 2 x 10.5 = 12.6
+11 to hit, 2 x (1d10+5) (Flurry of Blows) = 0.6 x 2 x 10.5 = 12.6
This all remains the same in the next 3 levels; 25.2 vs AC20.
Monk: Yes.
Monks can choose to use their attack's original damage or to use their Martial Arts damage dice. This often comes up since at lower levels, many monk weapons do higher damage than the Martial Arts dice can. As your quoted section says, monks can roll the Martial Arts damage dice instead of the normal damage.
Tavern Brawler: No.
Tavern brawler outright changes your damage dice for unarmed strike to 1d4. It doesn't have the language that Martial Arts does of allowing you to choose which damage amount to use.
Best Answer
Both tabaxi and tortles use the following phrasing:
Note that you can use your claws to make unarmed strikes; this means that if you want to, you still have the option of making normal unarmed strikes that deal 1 + your Strength modifier bludgeoning damage.
Now, the Monk:
The first bullet point is simple enough. You can use Dexterity instead of Strength for unarmed strikes. Note that, again, you can - you don't have to, so you can still choose to use Strength if you prefer.
The second bullet point says that you can roll a d4 in place of the normal damage of your unarmed strike. The question here is whether the "normal" damage of your unarmed strike is the default 1 + Str bludgeoning, or the 1d4 + Str slashing that your claws do. Ultimately, this will be up to your DM to decide. Personally, I'd say that your unarmed strikes "normally" deal 1d4 + Str slashing, so you should be able to replace the die with the Monk die. Importantly, it only replaces the die, so if your DM rules this way, you can still deal either slashing or bludgeoning as you choose.