IME, lightweight locking latchplates never really lock--they just hold the belt until the retractor kicks in. Frankly, they're kind of flimsy, but they do meet federal standards of lockability and some people never have a problem keeping their car seats tight. You seem to be having a problem with yours, so we have to figure this out. It would help a lot if you could post another picture showing the buckle side of the car seat as it's installed.
There are some tricks for dealing with this type of latchplate to keep it tight and they've been mentioned here. If the latchplate is too high up in the belt path and is sitting at an angle (like over the lockoff, for instance), then it won't stay locked. In this situation, twisting the buckle 2-3 full twists would shorten the buckle stalk and pull the latchplate down so that it lays flat against the belt. It's essential for the latchplate to lay flat against the belt to stay locked. If you can't twist the buckle, then flipping the latchplate over helps to keep it locked. Don't do both at the same time, though.
You can use both lockoffs, if necessary. It's easier to get the lockoff closed if you spread the seat belts out so they aren't exactly right on top of each other. That way there's not so much bulk and it will be easier to get the lockoff closed and to keep it closed. Keep in mind that if the seat belt is super tight, when you go to connect the tether and pull it tight, it will pull up on the seat belt (when you pull down at the top of the seat, the base wants to pull up against the seat belt) and that may make the lockoffs pop open as well. If this happens, loosen the seat belt a smidge.
When I install rear-facing seats, I like to stand behind them and lean over them rather than put my weight in them. That way I've got both my hands free to manipulate the seat belt and lockoffs.
HTH!