Yeah a poor soldier trying to cut up wooden pallets by bludeoning at them with a dull axe and stomping at them with the boots in a freezing winter night is something I expected to only see in cartoons instead of living it myself.
As OP seems to have figured out, you don’t have to play more than one round of firestorm really to unlock the axe unless you happen to drop at the same place as another guy and he kills you.
The ‘discussion’ about Firestorm here isn’t very constructive. It seems like most of your problems stem from the fact that it is a Battle Royale game, and the problems you have with it are present in any Battle Royale game. Which leads me to think that perhaps you just don’t like Battle Royale?
So, no one’s actually answered the question, whether you can buy the reward afterwards, or not. In the past, you’ve been able to buy the reward, if it was a weapon or vehicle, but not if it was a cosmetic item. Does anyone know if this is still the case?
That’s my understanding.
They were probably like… “but all that work warmed you up, right?” 
As for the topic, I hope it is and thought it would be later but NOT the overall chapter rewards. Don’t quote me on that, though. I’m currently on the “kill with reinforcements” part (because I’m also not a fan of Firestorm) but between being sick and not feeling the best and that being pretty situational and time consuming, I may not get it this week and wouldn’t mind getting the Axe later if I hear it’s good.
Oh I absolutely hate Battle Royale as a gamemode. There’s a reason I’m not at all interested in the likes of PUBG, Fortnite, Apex and so on. The thing is, I don’t have to play those at all. If I want to progress down the Tides of War though, I have to play atleast multiple games of Firestorm for the first objective before I can move onto the objectives where I don’t have to play Firestorm.
no you don’t, you can just buy the reward afterwards.
Yes, I do. I apologise for not stating it but I couldn’t care less about the actual reward/s at the end; it’s a freaking melee weapon, just how often am I actually going to use that (and likewise, the last two were face paint and a weapon skin, neither of which I will use since I don’t using face paint and I’m not big on the M30 Drilling nor do I like the look of the skin anyway). True, I don’t specifically have to, but you get the idea.
To progress down the tree, I first have to play enough Firestorm for 7,500xp in the mode. Yes, you can get ToW XP Just from playing any mode and slowly raising the ranks, but given the amount of time I get to actually sit down and play (mostly thanks to my brother and Destiny 2), I really appreciate the large jumps of ToW XP I can get from actually completing the objectives in the tree. Especially given how I can only look ahead to the next two weeks, it’s preferable to get good progress early on because I have no idea what’s to come later on in the chapter and the quicker I can reach Chapter Rank 40 So I can put BFV Away to collect dust again, the better.