The bits of grain and such are called "draff." Switching to a finer bag will help cut down the sediment. Letting the wort settle for 20-30min then decanting off the sediment before the boil will help too. In a "proper" MLT you can use the grain bed as a type of filter to clarify the wort.
I had a run of 7 or so astringent beers I traced back to getting too much draff into my boil kettle, so I'd say cutting down on the amount of draff is pretty important.
I used to do 12L BIAB batches exclusively when I lived in an apartment. I now have a 10 gallon round cooler MLT, but I'll still do BIAB if I'm doing a batch that's <12L or so. The thicker the grain bed, the better the filtration will be. So a round, tall cooler is better than a shallow, rectangular cooler, for a given volume. My 10 gallon cooler doesn't work so well for small batches.
When I moved into a bigger place I got all the goodies for 10 gallon batches: pump, propane burner, big pot, freezer, etc. The two things I couldn't live without now are my fermentation freezer and my MLT, so definitely get an MLT if you can. A 5 gallon beverage cooler would work fine for you.