I'm confused about having to pull the handle in order to lock it in to the base. I've seen many of this style of seat and never encountered needing to pull the handle to get it to pop in to the base. It may just be that it's stiff to pop on and pulling the handle makes it easier, but it definitely should not be necessary to pull the handle to pop the seat on to the base - just make sure you push down firmly on both ends of the seat when you place it on the base.
Is the kiddo a tall kid? Average size kids will usually last the seat until around 7 to 9 months depending on where height is proportioned (and that they're still under the weight limit.)
The other thing I would check once you have the snowsuit off, is that the harness doesn't have any excess sitting under the bum. The harness on this seat attaches on the back on one side of the seat, comes out the front and goes through the chest clip and buckle, then routes under the bum, comes back up on the opposite side, then goes through the buckle and chest clip, then out through the back of the shell. So if one tail is seeming long and the other short, it could be that the harness has become lopsided and you have room to let it out further.
These seats frequently have the covers not line up quite right, so while you're checking things out, double check that the straps are going through the same hole in the cover and the hard plastic shell of the seat - so if you're using top slot in the cover, it should be coming out the top slots at the back of the seat - if the straps are in different holes between the cover and shell, that can also account for loss of harness length.
I apologize if I'm stating something you've already noticed, but since it's a common mistake I see, I just wanted to also mention that there is a little diagram on the piece of plastic between the metal pieces that the harness straps route through. When you re-thread the harness strap ends through the metal piece, make sure that you're following that diagram. If the straps aren't routed properly, the harness won't stay at a fixed length properly.