No, the counters won't reset upon trading.
The whole step counting evolution method in general was super inconsistent with my several attempts to figure out how it works. It seems that sometimes, even if the required steps are taken, you may need to walk with the Pokemon a bit more to get it to evolve (I had several attempts where Rellor wouldn't evolve even though I leveled it up and it had 999+ steps). Generally as long as you don't walk with another Pokemon, the step counter won't reset. This means that you can shift the Pokemon's position in the party, put it in a box, shift it around in the box, and even trade the Pokemon, and the counter won't reset. I walked for 15 minutes with a Bramblin, traded it, leveled it up, and it evolved, and I did the same with a Rellor, although it didn't evolve immediately and I had to take a couple more steps (less than 999).
Primeape was thankfully straightforward, I used Rage Fist 10 times on the first game, used it 10 times on the second (finding a random shiny Flittle midway through), gave it a rare candy, and it evolved without a problem.
Since these are not trade evolution, Pawmo, Bramblin, Rellor, and Primeape will not evolve immediately after being traded with their requirements met. However, if you cancel the evolutions and still have the requirements met, you can just level the Pokemon up again and evolve them, even if you trade them again.
Hope this helped and I was able to brighten your day!
