A MSW will run the tools, but you need a PSW to recharge any batteries in a cordless. I believe my drill runs slower on MSW than on PSW, from how the water pump attachment acts, but I have no measurements of that.
Yes, you need at least #4 wire to the battery, but fatter is better. You can get those heavy duty, high amp parrot clamps. I use a cut off end of a set of heavy jumper cables that has those big clamps ISTR 600a or somesuch.
If you could do some cross-border shopping Can Tire is having another sale on their Xantrex built PSW 1000w. They just had a sale on these, so not sure what is going on there.
http://www.canadiantire.ca/en/pdp/motomaster-1000w-pure-sine-wave-inverter-w-remote-control-0111892p.html#.Uxo4fmeYZjo