The vast majority of yeast will go through a coffee filter let alone a strainer.. The easiest way to clear it is to just let it completely finish and settle, then syphon the 'clear' product off the top..
The trick to getting it to finish completely is to not add too much sugar..
If you add 3 cups to a gallon jug you're essentially adding enough sugar to bring the OG up to over 1.11, and the alcohol potential to over 17%.. You can buy yeasts that can survive this, but when you add the fermentable sugars from the strawberries even the champagne yeasts might be pushed beyond their finishing potential..
Baker's yeast has a finishing potential of about 14% alcohol.. If you try to surpass this, activity will dwindle off, and a few hardcore yeast cells will stay just active enough to keep it cloudy, and it will still taste like sweet.. (If you want more sweet, add it afterwards)..
Aside from that, I really like your recipe premise..
What size package of strawberries? Without them, if you add 825-850g (maybe 2 and a bit cups) of sugar to a gallon jug and top it up with some room left over, yeast will convert it to 14% alcohol.. And your strawberries are probably 20-30% fermentable by mass..
If you're trying to filter it then you have no idea how much nicer it will be when its cleared.. Once its finished (or dwindling) chilling it helps to knock the yeast down..
Lastly you might get slightly better flavor if you add some lemon juice, or cream of tartar (more neutral tasting).. Yeast like a pH aroung 4.7ish..