Yes absolutely I believe a bounce can go as high as $6K even.
And you are right that this dead cat bounce is going to cost a lot of people a lot of money. Because loads of people (n00bs) will fomo in and herald the coming of the next bull-run. This is why it will go up by quite a lot. The dead cat bounce is the final stage where they get completely rekt though. It will also destroy the last vestiges of people's belief in a recovery, and even moreso than today people will declare Bitcoin dead. It tried to regain it's heights, but topping out at $5-6K it failed to do so and will wither away and die, will be the general gist of the narrative. The truth is that it's simply far too early for any kind of bull run. Zoom out for a better perspective, only the monthly chart really matters when predicting the next bull run. And based on the monthly I think we are only halfway through the pain.

If you look at last crypto winter we bounced sideways between 200-300 for about a year. That's 50% fluctuations, but going nowhere. There's a big chance we will see some of this too.

You can try to profit off of the bounces though, I guess, if you're good enough. I'm fairly convinced that there is nothing to loose by stepping into a dead cat bounce and taking profits at 20+%. Just don't step into it with the plan of stepping out for a small profit.

Honestly I will NOT be really real bullish until we break $10-12K. And everything up until $8.5K is a potential dead cat bounce. For me, the upcoming period isn't so much about timing the start of the next bull run but rather bottom-fishing trying to get btc as cheap as possible. The bull run will inevitably come.
I mean, you could buy during a dead cat bounce and still come out on top if you hodl long enough... there's no real difference compared to buying 'at the bottom' or whatever. Eventually BTC will surpass it when the bull run comes. But psychologically it is a lot harder to buy a dead cat bounce, lose on it and keep hodling, than it is to just buy as cheap as possible and hodl.