A couple things.
First, if you have trouble breaking the weight off the floor but can lock out once it’s off the floor, it doesn’t really sound like a grip problem. There’s absolutely no reason that you can’t pull 400 with small hands (which I don’t think you really have). If you were talking about pulling 800, then a discussion of hand size may be more relevant.
Second, wrist wraps and wrist straps are two different things. Wraps are typically used to support the wrist in pressing or squatting movements and just wrap around the wrist. Some people use these on deadlifts to support the wrist, but I don’t imagine you get much of anything out of them. Wrist straps are wrapped around the bar in the opposite direction of you fingers. You then grab the strap. This absolutely helps grip and will typically help most people increase their deadlift.
Third, I’m 5’9" and measure about 8 inches from my wrist to the end of my middle finger. I wouldn’t say that I have a particularly strong grip, but I can handle 300lb farmer handles and have pulled 405 double overhand with chalk. I don’t think that grip is really your problem, but if it is, you should be able to overcome it.
My advice is to get chalk and then do more deadlifts double overhand.