Plastic case why? As long as you keep the batteries and uninsulated wire from touching the case it'll be fine.
As for keeping it tidy, you don't have to use the same wire to connect to the board and 510, I use solid 13G for the negative side and stranded for the 510 positive for strain relief. Then 28G for the signalling, I mean this board comes with 36ish gauge kynar wire, I left some of in on on my prototype, excuse the horrific positive wire soldering, I assembled the box in less than an hour to take on holiday!
Attach the battery negative to the switch, then the board negative from that contact, and pin 3 on the IC to the other side of the switch. In this you can see where most of the wires go, the black one holding the hot glue back connects to the dual 13Gs going to the MOSFET source pin. Someone told me I wouldn't even be able to get a MOSFET in there!
Here's a series one but disregard the wire colours, the wire up the left side (which goes into the switch and comes out red!) is actually the board's positive, the extra master switch is there just cause it fitted!
Wuwei.ap, that board looks cool, but the only reason I'm using this one is for the adjustable pulse width modulation really, straight up unregulated just doesn't cut it for me!
And that board you linked is gonna reduce the power output, it says 3.6V constant, so it's regulating the voltage down, by PWM, which means it'll switch your MOSFET on and off really quickly and give a lower average power!