Actually, it's the 1000 gallon filter pump, which still won't handle a bigger vacuum.

Could we attach a bigger pump to this size pool?

Here is a link that might be useful: Pool pump

We have an in-ground pool, so I am not sure on how the procedures differ, but I have to ask - when you say you tried shocking, do you mean you added some shock chemicals or did you do the process of shocking the pool until your overnight chlorine loss was less than .5 ppm?

If you are looking to learn more about defeating your algae, read up on the Pool School at the website for trouble free pool, it is extremely helpful. We just had to shock our pool (which is a process, not a one-time add of shock chemical) and we cleared our lime green pool in just over 48 hours.

My first thoughts are that it's not the pump, it's the shocking process, but my apologies if I'm overlooking something you've tried already. We'd had cloudy pool water for about a week, (I even broke down and added clarifier!) and my husband started saying "it's a problem with the sand filter! the pool store guy says we need new sand!" and I'd fought hubs on it, since most online forums say you pretty much don't ever need new sand. Woke up to a greenish tinge on the 8th day, so then, we followed shock procedures very closely. From Thurs night to Sat night, continuous testing and adding chlorine, and woke up to a sparkling clear pool. (Same filter! Same sand! ha ha!)

You need a bigger pump.