The base only was recalled, not the seat itself, and the OP stated she doesn't have the base anyway.
There are two BabyTrend seats, with several different bases. One seat is the fixed back, with harness slots, and you unthread and rethread the harness to move up. The other is the adjustable back, which has only one set of harness slots, and a knob in back that you turn to adjust the harness height. All of the pieces (bases and seats) are interchangeable, so if you wanted to get a new base, you could use it with your old seat.
Other than a version that was just released a month or two ago, BabyTrend seats have a 22 pound limit. There might be some with a 20 pound limit, so check yours.
My 10 month old, average height, 16 pound daughter still fits in hers with inches to grow.
It installs ok without the base. It's doable, but not something I'd want to do on a regular basis. I haven't needed noodles with it to install without the base. It's a bit of a pain to use that way, because if you put the kid in first, it's difficult to press down on the seat enough to get it tight during installation. If you install it first, the belt makes it difficult to get the kid in the seat, and then buckled. What I would do (and have done) is put kiddo in first, and then work around the child to get the seat installed.