I've looked at reports, and haven't seen anyone mention this regarding the ability to fish up pokemon.
If you leave the battle, and leave the pokemon alive; the pokemon just sit there, as if they are just a block placed there out of thin air.
Tested in various biomes, both vanilla biomes and BYG biomes.
Figured this is a bug, considering back in 1.12, the pokemon would swim around once you leave the battle.
Fished up Tympole floating near the surface of the water, unmoving; natural spawned Dewpider in the background, moving.

Tympole still unmoved, whilst a natural spawned Goldeen and Dewpider moves.

Mods also used: (Posted here, as the 'optional field' errored out when trying to submit)
Java version 8 update 251 (1.8.0_251-b08)
BYG-1.3.5, ExplorersCompass-1.16.5-1.1.2-forge
Hwyla-forge-1.10.11-B78_1.16.2
jei-1.16.5-7.7.1.152
journeymap-1.16.5-5.8.5p2
MouseTweaks-2.14-mc1.16.2
MyServerIsCompatible-1.16.5-1.0
NaturesCompass-1.16.5-1.9.1-forge
noteenoughcrashes-4.1.4+1.16.5-forge
OptiFine_1.16.5_HD_U_G8
Pixelmon-1.16.5-9.02-universal
PixelmonInformation-1.16.5-9.0.0-2.0.0