You could just break the groupe and specialize into blades and unarmed, it's still cheaper than taking clubs too. A group is better than separte skills if you want to use 3 skills at last. If you just need two, don't take the group.
I think not taking unarmed combat is just too bad because with such a character it already allows you to do great damage (maybe even more than with an axe, if you take the good options), and you never have to worry about carrying your fists. But if you really want an axe, even if it's useless, I can understand, I make too silly choices even on my recent characters just because "it has to be".
But if the axe is the only blade which you wanna use, then specialise… you can take blades 2 and specialise into axe, it'll make blades 2 axe 4, so you're at the same level than now, but it's cheaper in BP. For example : blades (axe) 2 and unarmed combat (boxing) 2 will allow you exactly the same dice pools than your melee group : 4 with the weapons you've got, but it costs 20 points instead of 40.