Those clips do come off, and in many designs you'd be able to thread them back on from the end. It is possible to clip them back on though.
The way I've found works is to gently prise open one end of the slot by twisting a screwdriver in it. Then slide that end over the piping, so that the piping replaces the screwdriver in the slot. The rest of the slot will go fairly easily.
Try to avoid doing this too often - if it keeps going you might be over-tightenting the chest strap, or snagging it on things.
It's also possible to replace the strap with one that wraps round the shoulder straps, though on that pack it wouldn't stay in place without further fastening