I may not be higher elo than you, but I hope my opinion will help
I main Amumu on TT, and so far it has worked great. I agree, his early is relatively weak, but I dont find the need to build very tanky or offensive. I usually start with boots and a ruby crystal, which allows me to be fast enough to escape and tanky enough to jungle on my own (with E). This allows your lanes to play solo and get more exp. By this point, if your bot is laning 2v1, he will either lose some farm or get harassed, so as soon as I hit lvl2 I try a gank with bandage toss. This usually gets a kill or makes the enemy fall back! After that, I usually alternate between doing jungle, trying a gank, and invading enemy jungle with my team.
An item I find a must-have on Amu is the hextech sweeper: AP and move speed are perfect for him, and the vision active is amazing! The cooldown reduction helps get Amu's ult quicker, and the hp makes gives him some more survivability. Thats pretty much the only core item.
The reason why I usually dont have to ponder about wether going more AP or tanky afterwards is because of the team compo. Since I am the only AP, and Amumu doesnt have as much burst, so I rarely get focused. The enemy team is then forced to make a hard choice:
- Take armor to protect from my 2 allies, but then get shreded by my damage
- Take magic resist to protect from me, but then my 2 allies destroy them with high AD
- Take health to try and mitigate both damages, but that only increases the damage my W deals.
Regardless of whether they take health or magic resist, Amu's passive combined with boots of sorcery and some masteries will allow him to dish out high amounts of damage.
PS: I play with relatively no defensive masteries (21/9/0) and no defensive runes.
TL;DR: In conclusion, yes Amumu is very viable in 3v3. To make up for his weak early, try to play with allies who are beefy and can survive the early combats pre-level 6 (like Hecarim, Xin, Vlad, etc). Playing him in lane is IMO a waste, so try to jungle him whenever possible. Having 2 AD champs + Amumu is often better since the enemies will have to decide between armor, health or magic resist, which gives you an advantage.