How can I do it as a fighter?
Is it easier to just be in heavier armor, but take fleet a bunch of times?
Well... as a fighter, you'll be able to move at full speed in heavy armor by 7th level. If you make it Mithral, then you can do it by 4th level. If I'm reading this right.
But I haven't slept in 30 hours so...
Fleet doesn't work in heavy armor for no reason at all. I hate that Fleet doesn't work unless you don't need it. : /
Mithral armor or boots of striding and springing are what is usually used to increase speed, so you may want to consider just wearing heavy armor.
If you want to use light armor to tank you need to heavily invest in dex. You're still going to be using a shield.
Light armor normal dex = like, 4 chain shirt, 2 dex. 16 AC.
Light armor huge dex shield = 4 chain shirt, 4 dex, 2 shield = 20 AC.
Then normal dex by like 11 is at 7 or so from chain shirt, 4 from miscellaneous bonuses and 3 from dex for 24.
Light armor huge dex shield is wearing like celestial chain mail for 9, then 6 from dex, a 4 shield and you're at 32.
We'll not even go into the fact that if your dex is huge then you're probably not doing much damage.
Free Hand Fighter has some advantages here, and going duelist allows you to add Int as well as Dex. In fact that's pretty much the best way to go for an unarmoured tank, unless you want to go unarmoured and dip a level of monk.
If you go 100% monk you can get an AC to die for, but it comes at a price of your offensive ability. It's not too bad if you want to go with maneuvers though.