The problem with splitting your aquisition team is that the capital will be able to take the worlds you send it at, but it will miss out on the experience given for killing off the pirates guarding the world.
I can't think of the map off hand, but one thing you could do is rush through each world and only killing the pirates in the world and neglecting to colonize the planet at all; taking a planet means that you have to spend resources to upgrade it that you could instead be increasing your fleet size. I would say definitely get the asteroid next to the homeworld since they are cheap to get, but move your fleet through to the world that is the "choke point" and colonize there.
By being able to block your opponent off into only going through 1 or 2 planets, you can leave your forces there as well as put up a lot of defense there, and stressing using repair bays; they can make a fleet of half the size almost as strong if not stronger due to the healing. Once you feel you have a large enough fleet at that planet, take the capital ship and go back over those other planets and take them and also upgrade them. This leaves your fleet short the capital, but the colonize bonus is just too great to use a colony frig, and you will most likely have level 2 or even level 3 colonize due to killing the pirates off.
Edit: ok looked at the map. best way to go about it is keep them limited to the planets they gain. have a scout stay in the grav well with the sun so you can see them coming at you from there, then colonize the asteroid and the ice planets, then just methodically move across the chain between you and them. my guess is they will send some LF, LRF, siege frigates, and a carrier cap through the sun grav well within the first 15 minutes at your homeworld, so you can either keep scouting the small 3 planet string between you and their homeworld to see what they have there, or be ready to fight in your homeworld. have at least 3 repair bays, all over lapping so they heal each other. scouting is important, so if you see that they have more fighters or bombers from carriers, get a few flak frigs to help with that.
I don't think it is all that hard once you have the repair bays and make the fight come to you. make sure your fleet fights in the range of the repair bays. also, if you let them get the desert planet in the middle, they have the upper hand on you, so be ready to fight hard to stay alive. remember getting the technology upgrades for damage are the least important for such a small map, so focus on adding to your fleet size or an armor upgrade.
Anything else, just ask.