So if you believe that relativity is true, then you must

believe that the front of ship B can be in two places at the same time!

Incorrect.

You have obviously had some exposure to relativity, but clearly have not fully understood it. This may be down to poor teaching or an unwillingness to learn. I'll give you the benefit of the doubt, but suggest that you go back to your teacher and explain that you haven't understood, because this would be a good example for class discussion.

It's a good example because it takes us back to the first principles of relativity. In your first or second lesson the teacher will have introduced you to the principle of simultaneity. The usual example, which I'm sure you can find on the net, is a train passing a platform.

On the front and back walls of the carriage are 2 clocks which the observer in the carriage wants to synchronise. This is done by setting off a flash bulb at the exact centre of the carriage and when the light reaches the clocks they start counting. So each clock is now showing the same time as the other, they are synchronised.

However, to the observer on the platform the light has the same speed as measured by the observer on the train, so they see the back wall moving towards the flash and the front wall moving away and so the light hits the clocks at different times. The observer on the platform says that the clocks do not show the same time, they are not synchronised.

(Note that this is very different from Galilean relativity where if 2 balls are thrown at the front & back walls, both observers will agree that they arrive at the same time. I'll leave you to work out why.)

This difference in time on the clocks is also why we see length contraction. If we measure the length of the train by using the speed of the train and timing when the front and back pass a point on the platform, then if we disagree on the time shown by the clocks we will disagree on the length of the train. For some reason many teachers teach time dilation and length contraction as 2 separate phenomena, I don't understand why because they are 2 aspects of the same thing.

If you understand the above you will understand almost all of special relativity (and quite o lot of GR as well), if you don't understand it discuss it with teacher and go over it until you do. This is important because most of the situations in relativity which puzzle people boil down to not understanding these basic principles.

Now you are in a good position to go back and solve your 'puzzle'. I won't do the arithmetic for you, but I'll give you another hint, synchronise the clocks when the noses of the 2 ships pass each other, then work out the times at the front and back of each ship for each observer for each event, you'll see that the problem disappears and your concept of 'at the same time' is faulty.

I'm assuming that you that you really do want to learn, so come back when you've done it and show us the results, otherwise don't bother posting any more silly 'puzzles' because they are all based on the same misunderstandings.