Here are a few process tricks that may help but no substitute for time. A nice starter will speed things along but I guess it's too late for that. If you are truly done fermenting, carefully rack it to a carboy and leave behind as much as you can. The racking will act as a way to resuspend some yeast to help clean things up flavor wise. A few days before you need to keg, chill the carboy and then hit it with gelatin. While clarity isn't an issue, the gelatin will help pull out some thin that would take longer to settle out.
When you rack to the keg, don't even get close to the bottom as to not stir things up. When you fill the keg, don't get greedy and top it up but leave a few inches head space. This will help out when you carbonate.
To carbonate, you need to know the temp of the beer to have any hope of not over shooting it. You could use a keg carbonation cap (which there are some tricks to) or what I find works best is to roll the keg making sure you have the gas tube up. Check one of the charts online to get the pressure for your volumes CO2 and don't cheat by going over it and hoping to go faster.
One last trick if you are worried about sediment would be to make a liquid to liquid jumper and then transfer to another keg. I don't think you need to since this one won't have the time to settle anything out.
Good luck and keep us posted.