I only saw a few minutes at the end, and was being rushed (no home internet), but in the last few moments you spent at cautious. Just an oversight, or intentional?
I was berserk up until the last few seconds. It always says "cautious" when you're berserk. After it wore off, I still didn't run because I was using the Chainsaw and running would have reduced my melee accuracy. If I had been using the shotgun, then yes, it would have been prudent for me to run and I did run for most of the game, but in this case, running would not have been to my advantage.
EDIT:
Having +80% fire resistance, the rocket blast would not knock you back more than 1~2 tiles.
I've heard people say a few times on this forum that fire resistance reduces knockback from rockets. I'm pretty sure that's untrue, so I decided to put it to the test. I just played an ITYTD speedrun in which I gathered the mods to make fireproof red armour, a power-modded rocket launcher, two envirosuits, and as many rockets as I could find.
Then, I rocket-jumped eighteen times without any armour or boots on, 0% fire resistance and 0% knockback reduction and counted how many squares I traveled. Here are the results:
15, 11, 9, 14, 8, 12, 12, 13, 16, 12, 12, 11, 10, 12, 8, 13, 12, 12; Average: 11.78
Then, I did the same with fireproof red armour on and envirosuit activated, 80% fire resistance and 0% knockback reduction and counted the squares for each rocket-jump again.
13, 12, 13, 12, 6, 15, 6, 10, 10, 10, 14, 11, 11, 15, 15, 13, 12, 16; Average: 11.89
The average numbers of squares traveled are very close. In fact, it was slightly higher when I had the higher fire resistance! (and certainly not 80% lower!) I think this test proves pretty conclusively that fire resistance does not reduce knockback from rocket explosions. I think it's safe to say that the game calculates knockback first and then damage reduction afterward. (Also, having the fire resistance made my rocket-jumps hurt a lot less; I was taking 1 or 2 points of damage at a time with the fire resistance, but without fire resistance, each jump knocked off about a third of my health!) I think it's also worth noting that on the occasions that I got knocked 16 squares, it activated berserk! (I was playing a melee build) Berserk got triggered whether I had resistances or not; this means it also calculates the "red screen flash" effect before taking damage reduction into account, which means you can get berserk triggered by getting hit without it doing much damage. (This has significant implications for melee games!)
Anyway, the point of all this is that using ranged weapons against Cybie still isn't a viable option for this challenge, because even though you aren't taking much damage from his rockets, the knockback will be throwing you around like a rag doll, and you won't get much of a chance to hit him back. I recognize that my chosen strategy of charging him with a Chainsaw is very luck dependant, but it's the only strategy that I can imagine working.