No
The Pokemon will not get the boost from Fisheous Rend/Bolt Beak
Here is the replay:
https://replay.pokemonshowdown.com/gen8doublesou-1174565272-kp0ymilwl4oafilusrlse1vssm5vsrrpw
So, in the replay, Incineroar uses Fake Out, causing Emolga to flinch and leaves it 81% HP. Dracovish, holding an Iron Ball for this, uses Fisheous Rend on Emolga, causing Emolga to have 21% HP.
Let's see what the calcs have to say:
For the calcs, Emolga is at 81% HP.
The calculations for Dracovish using Fisheous Rend when Speed EVs aren't inveseted:
252+ Atk Dracovish Fisheous Rend (85 BP) vs. /0 HP/ 0 Def Emolga: 180-213(71.7 - 84.8%) -- 31.3% chance to OHKO
Possible damage amounts: (180, 183, 184, 186, 189, 190, 193, 195, 198, 199, 201, 204, 205, 208, 210, 213)
Calcs for when Dracovish has Speed EVs invested:
252+ Atk Dracovish Fisheous Rend (170 BP) vs. 0 HP/ 0 Def Emolga: 358 - 423 (142.6 - 168.5%) -- gauranteed OHKO
Possible Damage Amounts: (358, 363, 367, 372, 375, 379, 384, 388, 393, 397, 400, 405, 409, 414, 418, 423)
To sum it up:
If Dracovish got the boost, its Fisheous Rend would've OHKOed Emolga, but it didn't. Dracovish could've OHKOed it without the boost, but the chance is 31.3%, so don't rely on that chance. So no, if a partner Pokemon uses Fake Out on a Pokemon faster than the Fossil Pokemon, it will not receive the Fisheous Rend/Bolt Beak boost.